1) Looking at the TIOBE index, all the static languages I recognised on there are: C,C++,C#,Visual Basic,Go,Fortran,Swift,Delphi,Cobol,Rust,Scala,Typescript,Kotlin,Haskell and D. Of these C and Go are the only two that don't appear to support generics so I don't think this approach excludes a lot of static languages.
2) If you want to distinguish GET and POST requests statically then you just need a type for them e.g.
GetRequest<TBody> implements Request<TBody> { }
if you don't need to do this then you can just add a method field and use a single type for both. Either way you don't need to use sum types so a language like Java can express it.
3) Yes you'll have to make functions that don't care about the body type generic so this approach could become unwieldy if you have a few such properties you want to track.
