It is all about time, the more time to defeat your security measures, the less likely the bad guys will hang around and risk getting caught.
a) does this merit putting up a couple of layers of plywood/sheetrock/metal for the walls, or should I just build it out "regular"?
If you are upgrading the door it makes sense to put in extra layers, look into expanded metal as a possible layer.
b) if I should go through the effort of hardening it, what kind of door should I use? I think a vault door is overkill...like a regular steel exterior door with longer hinge screws, etc be ok?
Vault door is overkill for what you are doing, steel door with steel frame, commercial is better if you can afford the extra costs. Good locks too (ASSA/Medeco, Mul-T-Lock, or equal).
Easy to DIY an alarm system for your room too (as part of the build).