I know quarters, paints, and appys can be. Can arabians and thoroughbreds be in the NCHA cutting competition too? What other breeds are allowed in this too?

It's not breed exclusive, it is competition where any horse of any breed or no particular breed is allowed to compete. If a horse has a talent, they can compete.

Just like other performance competitions like dressage, jumping, eventing, open rodeo.

It's on talent, not who the parents are .