It comes down to order of operations and how PHP type juggles. Hint: var_dump is your friend, echo always casts to a string so it is the worst operation for analyzing variable values, esp in debug settings.
Consider the following:
var_dump($a->$c); // array (size=1) / 'start' => int 2
var_dump($a->$c['start']); // array (size=1) / 'start' => int 2
var_dump($a->b['start']); // int 2
var_dump($c['start']); // string 'b' (length=1)
The key here is how PHP interprets the part of $c['start'] (include $c[0] here as well). $c is the string 'b', and when attempting to get the 'start' index of string 'b' this simply returns the first character in the string, which happens to simply be the only letter (b) in the string. You can test this out by using $c = 'bcdefg'; - it'll yield the same result (in this specific case). Also, $c['bogus'] will yield the exact same thing as $c['start']; food for thought, and make sure you do the required reading I linked to.
So with this in mind (knowing that $c['start'] reluctantly returns 'b'), the expression $a->$c['start'] is interpreted at $a->b. That is, the order is $a->($c['start']) and not ($a->$c)['start'].
Unfortunately you can't use () nor {} to steer the parser (PHP SCREAMs), so you won't be able to accomplish what you want in a single line. The following will work:
$e = 'start';
$interim = $a->$c;
echo $interim[$e];
Alternatively, you can cast your arrays as objects (if you have the luxury):
$a->$c = (object) $a->$c; // mutate
var_dump($a->$c->$e);
$interim = (object) $a->$c; // clone
var_dump($interim->$e);
...by the way, referring back up to $c['start'] and $c[0], in regards to $c[0][0] you simply can't do this because $c[0] is the character b in string 'b'; when access the character/byte b it will not have a property of [0].
var_dump()onprint $a->$c[0];to confirm it's array?