Are equalizers of regular functions always regular languages? (My guess is no because PCP, but...)
If you use Mealy machines, it forces your functions to be length-preserving, and therefore you cannot encode PCP with them.

Your regularity theorem holds with length-preserving functions.
If you want to allow length-increasing functions (that you need for PCP), you need a more powerful transducer model, for which undecidability quickly kicks in.