Ellie I'll try to answer your questions;
First you have the APHA, American Paint Horse Assoc., It is a breed all its own, they allow only Quarter horses in to the mix, but the Quarter Horse can be an appendix QH or have Thouroughbred in it, but the TB comes from the QH side. Lots of paints bred to QH's...lots.
Second you have the American Pinto horse Assoc. which is a color regristry (sp). Meaning any horse with enough white on it can be regristered with the Pinto Assoc. Arabs crosses, Paints, Saddlebreds, Mini's, etc.
If you get a solid colored or min. white on your Paint horse than it is still regristered as an American Paint Horse, but it's called a "Breeding Stock" Paint (which get very little respect in the paint world). But you can still breed them and get a colored baby (regristered).
The AQHA and the APHA are always changing thier rules on what can be regristered, how much white,etc., but for the most part, you can not have double regristered horse, but there are some out there that slipped through the loop holes one of the assoc. didn't close for a year or so. Both assoc. just changed the rules again in the beginning of 2004 so that you can't have double regristered horse.
Before 2004 a QH bred to a QH that produces a foal with too much white could be regristered under both QH and Paint assoc. That foal is called a "crop out" Quarter horse. again, I'm under the impression that the APHA changed that in 2004 to not allow crop outs to be regristered with them. QH assoc. has stated that crop outs are undesirable as did Paints, but as of 2003 they could be doubled regristered...how can that be undesirable? Again I have not kept up with the last rule change with crop outs but I believe that is how it is.
If you bred a paint to your Quarter Horse it could only be regristered under APHA (paint).
Hope I covered all your questions