I let the quilt sit overnight to dry before I begin the quilting. Depending on the size of the item and how much spray I have used I may or may not pin the edges. Usually, once I have done my stabilization stitching I then remove the pins and I have enough hold.
As for the cleaning of the carpet, maybe try a small spot with a little water and dish soap. If that works ok then proceed? I guess it depends on how big the area is that needs to be cleaned.