It doesn't need the entire codebase, it just needs the call map, the function signatures, etc. It doesn't have to include everything in a call - but having access to all of it means it can pick what seems relevant.

Yes, that's exactly right. The LLM gets a rough overview over the project (as you said, including function signatures and such) and will then decide what to open and use to complete/implement the objective.

In a real project the call map and function signatures are millions of tokens themselves.