What's a simple way to handle the ignition of flammable objects by spells?

Fun Question

For your specific points:

• Flammable is not the same as Vulnerable. Flammable means it can be set on fire without significant effort. Vulnerable indicates something especially damaging to the target. Wood-pulp paper is both flammable and Vulnerable to fire; humans are flammable but not Vulnerable; ice is not flammable but is Vulnerable to fire.
• No, fire spells should not (usually) have Advantage on attacks against flammable targets. Secondary effects do not make it easier to hit a target. The flammability is activated by appropriate sources of fire damage, not by hitting a specific AC. In the hypothetical case of a golem with a high AC from numerous layers of well-spaced paper, fire attacks would probably have Advantage, but that is a specific situation in which the nature of a flammable object would counteract AC, making certain attacks more likely to connect and cause damage.
• The mundane flask of oil (Player's Handbook, page 152, Oil) has guidelines for mundane fire. Taking 5 damage per round, until the fire burns out, is serious enough to be a threat but mild enough to not ruin a campaign. If you extend this rule to other flammable objects then you will need to house rule fitting durations.
• Yes, flammable applies at the DM's discretion. Common sense and consistency are important when making such rulings (else your players will become frustrated, angry, and / or despondent about interacting with flammables). It is also worthwhile to do some independent research for how flammable likely targets are; I mistakenly believed spider webs were flammable for a few years until I did research and some experimentation and learned that spider webs are not flammable.

My own experiences, on both sides of the DM screen, have shown me that one of the best ways to handle "accidental" combustion is to give the most inconvenienced parties a metagame resource, such as Inspiration, to reward the players for going along with the change. When the player characters are inconvenienced by the fire, this acknowledges the increased challenge and incentivizes them to deal with it. When the NPCs are inconvenienced by the fire, giving the NPCs Inspiration mitigates the effectiveness of the "burn everything" tactic in a way that lets the players succeed but still be aware that not all the advantages go to the player characters.

Good luck!

For simplicity, use a standard amount of damage

I'll answer three of your detail questions only in brief, as Vallhalla did a great job on those.

• Should flammable objects be vulnerable to fire damage? Often, but not generally: a dead tree may be flammable from his thin twigs and dry leaves, but not vulnerable due to its solid, massive trunk, for example.

• Should fire-damage spell attack rolls have advantage against flammable objects? No. Attack rolls are about how easy it is to hit, not what happens once you hit. That is what damage, resilience, vulnerability, resistance and immunity are for.

• Is the descriptor 'flammable' applied to objects at the DM's discretion? Yes. There is no list of what material or objects count as flammable or not, so it falls to the DM to decide, based on common English definition of the term as "easily set on fire".

• How should the ongoing damage of something being 'on fire' be adjudicated? As per the rules, you as a DM have to make this up entirely, and doing so on the spot is the opposite of easy. I want to explore this last aspect to make it easier when the situation comes up.

The approach I have been using successfully is: unless a specific rule tells me differently, objects take 1d6 fire damage per turn from being ignited. This is not realistic, but it fits to the design of 5e which optimizes game action over simulating real-world physics. You could also use 1d3 to stay under alchemist's fire, 1d4 as in alchemist's fire, 1d8, or 1d10 as the die unit of damage severity on p. 249 DMG. The reason I use d6 is part nostalgia from older editions, and part because it fits well to small to medium size objects for the typical fight.

A more realistic simple rule would be 1 fire damage per turn. However, for nearly any kind of combat situation this would be too slow to matter. The average combat in 5e takes about 3-4 rounds. At 1 damge per round, anything but tiny objects has no chance to be destroyed during combat, so this will not have an impact on the fight and can be attended to afterwards. From a fun with the game perspective it would not worth the effort to track it in that case (see Upper_case' answer for a more in-depth exploration of this consideration).

Realism

Let's take a look at how things burn in the real world for reference. For example, think of a barbeque: you set a small mound of charcoal on fire, using some accelerant to do so. The size of this amount of coal as an object is at best Small. Even if you considered them to be resilient and neither fragile nor vulnerable to fire damage, the coals have at best 3d6 hp.

How long does it take for those coals to be incinerated in the real world? In my experinece up to 2-3 hours. Even if you consider them destroyed long before they have been turned to ashes, they remain pretty solid for at least half an hour. That is 300 rounds of combat, or less than 1/20 of a point of damage per round. 1 fire damage is the minimum you could deal. Even at 1 fire damage per round, you would destroy them more than 20 times as fast as in reality.

There are of course other complications, like flames being small and weak licks intitially, and over time becoming an all-engulfing conflagration, which would suggest a non-linear damage development, which, like with falling damage, we'll ignore for simplicity.

Rules benchmarks

There are two mundane items in the rules that give us precedent for damage from igniting things

• Alchemist’s Fire. This sticky, adhesive fluid ignites when exposed to air. [...] On a hit, the target takes 1d4 fire damage at the start of each of its turns.

• Oil. [...] If the target takes any fire damage before the oil dries (after 1 minute), the target takes an additional 5 fire damage from the burning oil.

Both of them provide higher damage output than the minimum 1 fire damage; Alchemist's Fire is self-igniting and hence cheaper on the action economy to get going, and deals 1d4. Oil is more difficult to get started, and deals nearly the equivalent of 1d10. But both of them involve special helper substances¹ that are missing for normal ignition.

Setting something on fire with a spell would deal more damage than alchemist's fire, which may feel wrong. If that concerns you, maybe use 1d4 or 1d3 instead of 1d6.

¹ Interestingly, even though the volumes involved are certainly tiny, fragile and/or vulnerable to fire damage, indicating maybe 1d4 hp, they themselves do NOT burn away in one round, Alchemist Fire is without any limit on its duration, Oil is good for a minute. This touches upon the question if liquids even count as objects - their ability to burn is one of the factors supporting the pro side.

It's important to find the fun for your table before putting much effort into mechanics

A key observation in this case is that D&D 5e doesn't simulate much around causing or spreading fires in combat, which matters because there aren't very many mechanics around those things. As you've noted, the core rulebooks generally only go as far as indicating that an object could be lit on fire if it's flammable.

That's the first big consideration: detailed mechanics around how things burn (how easily they ignite, how quickly they are destroyed by fire, how fire might naturally spread around the area) don't really exist, and so neither do applications of those properties. So before working out a detailed system to apply those properties it's worth considering why it will be worth your (and your players') attention to interact with them. And with homebrewed mechanics that can get really table-specific and so it's a great opportunity to create something tailored to your players.

For example, if your players have fun preparing battlefields with hazards and like the idea of preparing burn zones mid-combat, then details like how easily or how quickly fire spreads from one object to another would be important to focus on and how much damage an object takes from being on fire may be less so. Then it's more a question about what types of objects PCs might be interested in moving around and burning and less about how durable they are to flames.

My players definitely like getting extra mileage out of their spells, including starting fires, but they are not so interested in very precise object interactions like burnability of random objects. So for my players it would be too fussy and niche-applicable to worry about those kinds of details, and it would degrade the game to spend time modeling and considering them all the time. We still do all kinds of fire-related stuff, but I fold it into the streamlined nature of 5e: things ignite, or don't, producing environmental hazards; interacting with burning objects grants Advantage (like when trying to break down a door) or Disadvantage (like when trying to do investigate the interior of a building that's on fire); fire damage translates directly to what a fire from Create Bonfire would do; and other things like that.

But that's us, and other tables might not have as strong a preference for ease and fluidity of play versus more number-crunching content.

I do not recommend developing and choosing mechanics in a vacuum, then seeing what happens; I recommend thinking about what you want to be possible (but currently isn't) and then create new mechanics to make it possible.

In games that I run, my players try to get all kinds of extra benefits out of spell effects. That very much includes fire and heat. But what I have discovered my players most want are, in roughly this order, to:

1. Think of clever solutions to problems that use their characters' special abilities
2. Overcome in-game obstacles more easily than what they assume the intended approaches would allow
3. Do things with some extra dramatic flair

I'll give two examples of fire spells in this context from a game I'm running now.

Case 1:

In the first case, the PCs were sneaking into a building to steal some sensitive documents and discovered a prisoner caged in the office they were ransacking. My most fire-happy player decided that they should steal the documents and free the prisoner, then use a fire to destroy the evidence. I ruled that it worked, but that the fire quickly spread out of control and engulfed the entire building. The flammability of the building created a narrative situation in which they had an effective time limit to escape alive, and with certain routes blocked off due to a new environmental hazard. Plus lots of complications after the fact.

In this case, more specificity wouldn't have made the encounter more interesting but definitely would have multiplied the amount of work I had to do. Tracking the flammability of the wooden desk vs. the wooden floor vs. the bookshelves would have added a lot of bookkeeping, and what I would have gained from it would have been a more slowly-spreading fire (and therefore fewer complications to the encounter) and a more formulaically-spreading fire. More predictability and less peril is not what I was going for. Even if I had had house rules for flammability of all objects in place, I would have thrown them out for this sequence.

What I ended up choosing interacted with elements 1 and 3 from my list: fire magic made hiding evidence easier, and was certainly dramatic. But while the player expected to get 2 as well (no evidence, we'll be totally in the clear!), they got the opposite. Which also made sense in-game, as that character had no expertise in forensic fire investigation and should not be able to casually predict exactly how a flammable building will ignite.

Case 2:

The same party in the same game came upon a locked vault door with very little time to break in but a strong interest in doing so. They had not prepared enough to get into the vault by any means I had planned, and a vault door that can be casually bypassed at any time may as well not be there at all. They learned through investigation that the most vulnerable part of the vault door was a collection of hinge mechanisms that would be very hard to destroy in the time they had available. A player came up with the idea of using magical fire to heat the hinge mechanisms and make them easier to break.

That's not too realistic, but I ultimately said yes: PCs attacking a hinge mechanism in the middle of a bonfire got to attack with Advantage. It was a novel use of the PC's magical abilities (item 1), helped them deal with an obstacle in a novel way (item 2, and especially important as they had no other way of getting in), and was dramatic (item 3). It doesn't matter that thick metal objects aren't flammable, or that I didn't have a full suite of mechanics dealing with heating metals. The party got to solve their immediate problem with creative use of the resources at their disposal. The mechanics of how that played out were the least important part of what was a tense and fun encounter.

Regarding the specific questions posted in the OP:

• Should flammable objects be vulnerable to fire damage?

I say that this one is largely irrelevant. Objects being destroyed or not by the end of combat usually doesn't matter at all, and consequently neither does how quickly or easily an object is destroyed. In cases where this does matter (and you could definitely engineer as many of these as you like), the challenge is almost certainly going to come from issues that surround the object's destruction. Those are in your hands as DM, and creating the scenario will require you to account for this one way or another.

As an example, if your players need to destroy a flammable object within 10 turns to accomplish their objective, and vulnerability to fire would make that too easy to do, then you would either need to increase the object's HP, introduce other obstacles which prevent the PCs from igniting it too early in the combat (or something similar), or choose to use a non-flammable object instead. Any such mitigation would just make the vulnerability irrelevant. So this idea, to me, just feels like mechanic and bookkeeping bloat.

• Should fire-damage spell attack rolls have advantage against flammable objects?

As with the previous bullet point, I don't think that it matters. But if pressed to it I'd say no. An attack roll for a fire spell is usually to see if the spell itself hits in a meaningful way, not to see if the target ignites. If you miss with Firebolt, the bolt of fire doesn't hit what you were trying to hit. Being flammable alone doesn't make a target easier to hit, only easier to set ablaze if it is hit.

If you have flammability as an option with meaningful gameplay benefits and a fire-happy table of players, you are already in for a lot of fires. Granting Advantage to efforts to start fires will exacerbate this while also making those benefits easier to gain. But if you're ready for those, then I don't think that this one would matter much either way as it would be subsumed by encounter design concerns.

• How should the ongoing damage of something being 'on fire' be adjudicated?

I, personally, just treat burning objects as though they were in the middle of a fire created by the most basic casting of the Create Bonfire spell but without the opportunity for a Saving Throw. If something is on fire, it can't dodge the heat or flames! So it's generally 1d8 per turn.

• Is the descriptor 'flammable' applied to objects at the DM's discretion?

Of course. The word flammable itself is a kind-of-subjective term: many things will burn in the right conditions, but most people are concerned with the conditions they are most likely to encounter. Water isn't generally thought of as flammable, but hurl a chunk of cesium into it and you'll see it "burn". Wax candles are designed to burn in everyday conditions, but a candle in a vacuum will never ignite in candles' usual way.

The importance of whether or not an object is described as flammable can only be evaluated in the context of how important you have decided a burning object will be. But I do not think that a DM adding or removing that description from objects will, in itself, have much material effect on the game.

I think simple is best. The DM can decide what items are affected by the spell, and which of those items are flammable.

If it is a medium or smaller flammable object, I usually just treat it as if a flask of oil was poured and lit there. Maybe have it last only 1 round for a tiny object, or longer if it is more substantial (e.g. a wooden chest).

Larger objects may be only partially on fire (flames in one 5' square), or if it was an AoE spell like Fireball, the whole thing might now be a flaming hazard.