When I paper piece, I use regular copy paper which I have purchased from Walmart. I make sure to get the least expensive paper I find, not the top-quality bond paper I use for typing papers, etc. I have never had any trouble with it. Whatever needle is in my machine is the one I use for paper piecing. It's usually a 12 or 14 since that is the size I use most.
I will say that I put in a new needle when I begin a new project, but I paper piece with the same needle that is in the machine. The needle will dull fairly quickly with paper piecing, so using an old needle won't make that much difference. I'll paper piece for a while then replace with a new needle if I'm doing a new "regular" project. I hope that makes sense.
I have been told that using newsprint or vellum will produce better results, but I don't want to go against what already works for me so well. If you've found a method which works for you, then by all means continue to use it and enjoy the process!