re #1: If you are talking about functionality, the answer is no. The difference between self-hosted and cloud editions is the operational and availability burdens. In the former case, it is on you and your team, in the latter it is on FusionAuth's team.
re #2: yes, they are equivalent at similar editions. So if you have community edition, it is the same code in both the cloud and the self-hosted situations (for both validating the id token and anything else). Same is true for starter, essentials or enterprise.
Hope this helps.