If I have a piece of equipment rated for a Class 1 area...can I install in Class 2 area?
Not necessarily, they each have unique requirements. See Section 500.8(B)(1) [2008 NEC] BTW, I don't usually make a big deal of it but the way the question was asked leads me to note that "Class" is designated with Roman numerals like "I" or "II." "Division" is designated with Arabic numerals.
For motors, is it permissible to swap junction boxes if the motors are identical (same manufacturer and model)?
Are you asking about motors in general or Hazloc applications?
No I don't think so. A piece of equipment rated Class I Division I can usually be installed in a Class I Division II but that's not what you have here. Class I is for combustible/flammable gases and vapors. Class II is for combustible dusts.