The Pinto horse registry if your horse is a pinto. Half Arab if the registered horse is an Arabian. And I believe the appendix association as long as one of the registered parents is a Quarter Horse, and you can get a blood draw to get the DNA of the foal to prove that the registered parent is really the parent. Know someone who got a "feed lot" horse, dam was a Quarter Horse, have no idea what dad was, but they were able to get her into the Appendix registry. I think it also depends on how much effort you want to put into it. Some will take a horse no problem, so long as you have the registered parent's papers ect. others need paperwork, DNA tests, ect.
I'm not sure where this information comes from but it's not completely true. The Appendix Association for QH needs the other parent to be a TB, not just any horse. They can be registered with the AQHA as an appendix also but not if one parent is TWH or a Morgan or anything other then a TB.
As for DNA, you still need to know who the parent is so they can cross refernece the DNA they have on file - IF they have it on file. They will not take your DNA and try to find it in their data bank. Even if you know who the parents are and the DNA matches, it still isn't that easy. You have to get papers signed by the stallion owner and the dam owner, then, depending on how old the horse is, it can get very, very costly.
As far as that feedlot horse being registered as an appendix - no. Not without proof of both parents and the stallion had to be a TB. Then they had to get papers signed by the dam's owner and the stallion's owner - and pay a good deal of money ... all doubtful.
This is the kind of question that an owner of a grade horse would ask thining that they can get papers on him/her.