Do I pin the pattern to the fabric to cut out the pieces?
Cut the pattern pieces on the cutting line first (this will be marked on the paper). Then pin it to the fabric and cut out the pieces. Pay attention to the grainline marked on the pattern (this will be a long arrow usually in the middle of the piece).
If so, after I've got the pieces cut out, do I leave the pattern pinned to it?
Yes, until you are ready to use the piece. Keeps you from forgetting what piece is what.
If yes, do I sew the pieces together with the fabric paper still attached?
No, take the paper off. Pin the fabric pieces together but don't sew the pattern paper to them.
What if I want to reuse the pattern? Do I need to make a copy on tracing paper first?
No, you don't need to make a copy. I usually fold them up and put the pieces and the envelope they came in into a ziploc bag. No matter how hard you try, those darn pieces are never going to fit neatly back in that darn pattern envelope.