If you have an AQHA Appendix mare bred to a AQHA stud, could the foal be fully registered as a full AQHA? Or would it be only as an AQHA Appendix?

The foal would be appendix also. I will add this though. You can do everything with an appendix horse that you can do with a horse the is permanent registered((regular registry). The only thing that you cannot do is BREED to a thoroughbred or another appendix horse. Is that understandable?  My neighbor had a son of Dash For Perks that was appendix and he bred his own mares(all permanent registered) to him for years. All offspring was appendix. There are a few ways that one can be changed to permanent registered. You can show in judged events, race, run in timed events anything with appendix. I have had several. 

Could the appendix ever be bred out of the papers? Or not? And they probably wouldn't sell for as much, would they?

An appendix can earn their regular registry by obtaining their ROM through AQHA shows or racing. 

But as it pertains to barrel racing, they would just stay appendix?

So thankful for all the info. Never dealt with appendixes before

They can earn a ROM in barrel racing after earning 10 points running AQHA shows. 
I'm actually currently preparing to try and do this with one of my mares who has appendix papers. My long term goal is to earn her regular papers. 
Otherwise you just have to be mindful when breeding an appendix. You can only breed to regular registry QH.

Two ways they can be changed to regular registry. One is to have a speed index above 80. (Which gives them a racing register of merit. The other way is to have a performance register of merit which means they have to have competed in some AQHA event at approved AQHA shows. They have to accumulate 15 points to do this. My sweet old broodmare had a speed index of 92 and no one ever bothered to have her papers changed until she was 16. I bought a horse that was about 8 and she was appendix and I had her papers changed after I bought her. My neighbor bought a very nice stallion that was injured as a youngster and had no hope of getting a ROM.  It doesn't make any difference if they are appendix especially if you are going to race or barrel race. Agai, I hope I explained that well. When I can't sleep at night, I read the AQHA RULEBOOK, that will definitely put you to sleep!!  

If you get at least 10 points (thought15 but checked, oops) in barrel racing they can be changed. It will not make any difference if you barrel race with an appendix horse or permanent registered one whether in an open barrel race or AQHA. The only place that appendix matters is when you breed. That's it, I promise!  

In order to get points in AQHA classes they go by number of entries. It is very hard to get points because unless you go to big shows there are not enough entries to get event a point. Will try to find the point scale. 

Here is a link to several point systems. AQHA is the top one. 

https://www.showhorsepromotions.com/points.htm