#1
I'm not quite sure if this is it, but I'm looking for a keyboard that I can use with PT9 as a midi controller, and also store the patches I make on the keyboard itself so I can have the patches wherever I go. Is mapping what that is?

and if not, where can I find a keyboard that does it? (if there is one)
Check out my music, if you please.
#3
No, that's not mapping. Mapping is setting up various MIDI parameters to be controlled by various things on the controller, like knobs, sliders, the pitch bend wheel, the mod wheel, aftertouch, velocity, etc.

You want a synth of some sort. Most of them can be used as MIDI keyboards in addition to playing their own sounds, although I don't think you can use mapping, since the knobs on the synth already have dedicated functions.