Re: regularization - by using fewer numbers to represent the same output info, you are implicitly reducing the dimensionality of your function approximate.

Re: (a), (b) Generally in big nets, you want to regularize because you will otherwise overfit. It’s not about the output dimension, it’s that you have a giant approximator (ie a billion params) fitting a much smaller data dimensionality and you have to do something about that. The output can be “cat or not” and you’ll still have the same problem.