Problems 1 (ValueError from help(os)) and 2 ('module' as first param) were simple fixes:
The attached patch fixes problem 1 (and is probably worth doing anyway, since inspect.signature has the ability to raise ValueError, and (IMO) help(<module>) should never raise an exception.
Problem 2 is just a matter of adding self.show_in_signature = False to the first param of module-level functions. I've left a review comment at the right line.
Problem 3 would also be fixed by re-adding 'module' to c_keywords, but since you want it to work without that, that's out, and I'm not sure what the proper fix would be otherwise.
The current patch no longer applies cleanly after #20287; I would just post an updated version of the entire patch with my changes as well, but the merge is not trivial and I don't want to screw it up :)