Regular romex should be fine...just like anything else, that is supposed to be a dry area. The wet stuff is required if it is expected to get wet.
I think you'll be happier if the gfci is in the panel since it would not be a major pain to get to if it ever did trip. Plus, you are supposed to check it on a periodic basis which would never happen if it was burried beneath the tub.
Just make sure that the switch you select is rated for the 20A supply, and it should be fine. Most wall switches are only rated for a 15A load, so you may have to go to an electrical supply...depends on where you are looking.