yes, the only "sport" which requires an animal to be intact and officially fully registered with the AKC is the breed ring. altered AKC registered animals can compete in any other sport that their breed allows.
as for rescues, if you apply for an ILP, your dog can compete if it looks fairly purebred and is spayed/neutered - THIS IS KEY. they HAVE to be spayed/neutered.
my own rescue competes with his ILP.