When I paper piece, I do a quick finger press to make sure my pieces are large enough. If everything checks out, I cut off the excess, save it and then press it open w/ the iron.
I don't remove the paper until I am ready to sew everything together. Unfortunately, I have many projects in process that are months old--it lets me see what I'm doing and keeps my edges neater.
HTH--looks like you are well on your way w/ the FG
Julie