> Does C also have a compiler that turns C code into assembly

Yes, that's 'a C compiler', like gcc.

> before the real runtime does its work

Sort of, the program is 'the runtime', but this is backwards, languages that have 'a runtime' get the name from it running at runtime to compile/interpret source or byte code. In C what runs at runtime is just your program, whatever you compiled. (Maybe it's an interpreter though!)