Sorry for the long answer. I have split it into sections for readability.
Regarding the Background Premise
To begin, I don’t actually think this question helps us get at what makes us human or what distinguishes a human from an animal (or, since humans are animals, from a non-human animal). The idea that an answer would be revealing in this way rests on an old (and common) Aristotelian assumption that whatever is unique about humanity is the essence of humanity. But while unique features are useful way of tracking a kind of thing we are looking for, they are not necessarily definitions.
An example: if we were in the Gobi desert and I wanted you to bring me an antelope, I could say “bring me one of the four-legged, horned animals” (assuming you didn’t already know what an antelope was). But “an animal with four legs and horns” is not the definition of an antelope. It’s just a serviceable way of tracking what I want in situation we are in (since there are no other four-legged, horned animals in the Gobi desert). So picking out an antelope (or a human) does not necessarily require knowing the defining features of being an antelope (or a human).
In any case, “human” is ultimately a taxonomical category. Humans are defined by their biology, not their abilities. Indeed, our concern with abilities usually comes in not when we are asking whether or not something is human, but whether it is a person. And while it is a common error to conflate the two categories, they are most certainly different. A human that lacks linguistic or rational capacities is still a human. It’s just not the kind of being we think exemplifies humanity. So now let’s look at the various ways you suggested we might approach the question.
(1) If a human were turned into an animal, (a.k.a. “Bewitched”, “Jeanie” or little Anthony Fremont but in a less isolated setting)—how could a person get their human rights recognized or restored?
Assuming that the (former) human retains its memories, the most straightforward way would probably be to do something that the type of animal it has been turned into cannot do. Scratching out a condensed version of what happened on an appropriate surface—wood, sand, whatever—would probably go a long way to getting recognized (something like, “My name is Joe, and I have been turned into a cat by a witch. What else can I do to convince you?”) Of course, this depends on the capacities of the animal you have been turned into. A human-turned-fish might have a significantly harder time than a human-turned-cat, for example.
—Humanlike animals, such as monkeys
A humanlike animal could presumably never prove that it is a human because it is not. Again, “human” is a taxonomical category. Whether or not they could prove themselves to be persons is another question altogether. Not only would they need an answer to the ongoing debate over what constitutes personhood, they would need a concept of personhood in the first place and a desire to prove themselves persons.
—Common, likeable domesticated animals—dogs, cats, horses, birds—some of which (birds) might retain the ability to speak, or otherwise understand and communicate
Same as above.
—more exotic animals such as an elephant or rabbit or snake
Same as above.
—something monstrous or (1e)—truly disgusting and repulsive but not human or even a known or identifiable animal
Same as above insofar as we are starting with the premise that they are not taxonomically human. That said, plenty of actual humans have been considered monstrous, disgusting, and/or repulsive by others in the past. Here we can only appeal to the fact that appearance is not a moral quality and that disgust is not a reliable indicator of any sort of value other than aesthetic.
(2) What if a person (seriously—because of a medical condition or transformation/transmutation) LOOKED very much like some other primate ??? What other than speech might prove their humanity?
A DNA test.
(3) (the most serious of the three possibilities mentioned here) If a Chimp, Gorilla, or other primate COULD obtain human or near-human sentience, could or would they be allowed to pass as human in society?
“Passing” is not a matter of being allowed. It is a matter of what other people actually think. For a primate to pass as human, it would have to appear human to passersby. That’s what passing is. If you mean to ask whether it would be allowed to contribute to or participate in society the way we allow humans to do, I think we probably would. However, the novelty of it all would probably limit it to certain careers at first (since some industries would be more interested in it than others).
(4) I’ll throw this in, too—the movie “A.I.” brought this up—what about androids and machines—could they ever feel human emotions and/or be recognized as human?
Once again, they would never actually be human because “human” is a taxonomical category. So if we recognized them as such, we would be mistaken. That said, humans are just complicated biological machines. As such, there does not seem to be any a priori reason for thinking that we will never be able to produce a machine that has all of the same capacities as a human being (including the capacity to feel the same emotions that humans feel). Whether or not we’ll get there before we go extinct, however, is a question that only the unfolding of history can reveal.