Yes it does. For example: "para butów" (a pair of shoes). But you should be careful with that word, because there is a very similar one "parę". It does not change and it means "a few". "Mam parę ręczników." (I have a few towels.) "Para ręczników jest w łazience." (A pair of towels is in the bathroom.)
Yes, it's Instrumental. The word "para" (couple) is feminine no matter who is a part of this couple.
I still can't believe that English can use "pair" about people, but apparently it can. OK, added.