I believe if you want to register her (wtih AQHA) you'll need to know her dam and sire and possibly even their registration numbers.
I'm not sure if there's another registry you can register with, but if there is, I'd want to know. I have a QH gelding and I don't have his papers (doubt I ever will) and I would just like him to be registered.
As for showing, you can show her, it just couldn't be any breed shows I believe.
Registering her really depends on her parents and whether or not they are registered or purebreds. As for showing, you can show any horse you want. You run into issues if you enter certain hunter A classes but even at that point there are ways to qualify and enter those anyways.
Yeah, the AQHA will not register a horse without proof of lineage and they also require a dna typing I think. Too many people claiming their horses were bred a certain way when they weren't. You can find open shows that will accept any breed or you could go for a specific discipline and show there; NRHA, NBRA, etc. I don't think they require papers but I'm not sure.