To answer your first question, filling out the form, you will need to get these details from the Azure AD you setup. The Domain Name is the domain you created in Azure Ad. The Application ID is the guid you got when you registered your application in your Azure AD. The last field, Sign-up or sign-in policy, is the Azure AD policy you want to use to manage people signing up for your service as well as signing into your service.
The problem is, setting up your Azure AD is only one step out of many. What you should be learning how to do is setting up Single Sign On (SSO) using Azure AD. For that, I suggest looking at Authentication Scenarios for Azure AD, What is application access and single sign-on with Azure Active Directory? and Azure Active Directory B2C: Built-in policies. These series of articles should put you on the right path to get started with using Azure AD.
Your second question can be subjective, so I'll simply point things you will need to concern yourself with if you try to implement your own authentication. The biggest problem with doing it your self is making sure you have addressed the necessary security concerns. You will need to have your passwords stored securely which means salting and hashing them (I suggest Googling if you aren't aware of those term). You will also need to handle scenarios like password reset, forgetting user name and/or handling inactive or disabled user accounts. Many organizations and developers like using third party providers for SSO so they don't have to deal with such issues.
For your last question, yes there are. Microsoft does include a basic one with their web project templates (if you choose) and there are other providers out there such as Google or Facebook. There are many other options out there that are open source. A quick search on NuGet yielded over 2k results (https://www.nuget.org/packages?q=user+authentication).
Web Application (Model View Controller)) says that it can also be used for restful services (i.e. it's described the opposite way around from the template you've picked) and that template does offer local accounts.