Right. This is where math-oriented people get stuck on counting half-steps and forget that the assigned names carry information. E-B# is a different thing from E-C which is different from abominations like E-Dbb.

This is off. E-C# is the major sixth. E-C is the minor sixth, which as you say is enharmonic with sharp fifth (E-B#).

Yes, how could I gaffe so badly? Inverting a major third gives a minor sixth. Rule of nine, minor-major swap. Elementary school stuff.