Not really in the case of TypeScript, because (with very small exceptions) when you “compile” TypeScript you are literally just removing the TypeScript, leaving plain JavaScript. It’s just type annotations; it doesn’t describe any runtime behavior at all.
That depends on both the target and the typescript features you use. In many cases, even when down leveling isn't involved, transpiled code can result in more than just stripping type info (particularly common in classes or things with helper functions). There's also nothing stopping a typescript compiler from optimizing transpiled (or directly compiled) code like any other compiler would, though the default typescript tools don't really go after any of that (or even produce a minified version itself using the additional type hints).
Agreed, at least usually right now (it doesn't have to be forever, which would probably be the most realistic way for TypeScript to make meaningful runtime gains). That does not preclude the possibility of producing more optimal JavaScript code for the runtime to consume. I give a couple examples of that in the other comments.
