So since every(?) class can use the NS PDWs, will buying it on one class let me use it on the others? For example, if I buy one on a Medic, can I use it on a Heavy Asssault?
Yes, and it's the same for all weapons that can be used by more than one class. You unlock it once and all classes that can use said weapon can use it.
Yes, and if you buy NS weapons with station cash, you can use them on any character you have now or ever make.