Since we are in the fediverse, I think it would be convenient to have a fediverse-wide resolvable fediverse URI scheme, that would look like so:
fediverse://
edit: Found a relevant FEP: https://codeberg.org/fediverse/fep/src/branch/main/fep/07d7/fep-07d7.md
Absolutely not. It should run on HTTP, as a website. Unless you want to build a client which would be somehow fundamentally different from a web browser somehow (note: Lagrange and Gopher Browser are just browsers), which would somehow be able to display data from every use of ActivityPub / “the fediverse” in a different context from a web browser, then no. What we need to build is website software more in line with kbin / mbin, collecting together all the different information of the fediverse into one interface.