First, we need to ask if are we talking about moral rights or legal rights. Why does this matter? For one, legal personhood is fairly well defined in the American and European legal systems. A legal person is a bearer of legal rights and/or protections, and legal personhood rests on the ability of an entity to assert said rights or protections (that ability being itself defined as the ability to be party to a legal dispute).
Who and/or what counts as a legal person is therefore defined—albeit diffusely—by statute and can be expanded or retracted entirely at the whim of legislative bodies. Assuming that judges are operating within the law, they cannot declare something a legal person that is not already (directly or indirectly) defined as such (which would suggest that the judges in this case made the legally right decision assuming they did not miss some umbra of the law under which an elephant should count as a legal person).
If you are completely dissatisfied with that reading, however, you aren’t alone. Few people are truly concerned with mere legal personhood. The issue for them is moral personhood and the moral rights that attach to it. For while some people see morality and legality is entirely separate endeavors, most people see legal rights as being constrained by moral rights (meaning that one cannot legally deny someone anything to which they have a moral right).
But this is where things get complicated because the notion of animal rights comes five or six steps down the chain of ethical reasoning, which means that one has typically committed to a number of stances before the issue can even be addressed. Thus one has to do a lot of backtracking to really understand and respond to another person’s view on the subject. To discuss an animal’s moral rights, we need to answer more basic questions about the source of morality, the definition of a moral person, and the nature of rights.
That said, I do think we can answer some of your questions while remaining agnostic about those issues. First, you ask which animal rights make sense. Here we can go all the way back to Aristotle and note that, at the very least, what we owe to any given entity depends on the sort of thing that it is. It would make no sense, for example, to say that I have the right to photosynthesize. Similarly, it makes no sense to say that an ant has a right to a college education. There are many things this does not rule out, but it does support the notion that the list of human rights and the list of animal rights is not coextensive. (Unmentioned so far, however, is that it also opens up the possibility that animals might have rights that humans do not.)
I think we can also answer the question of whether or not animal rights are subservient to human rights on purely conceptual grounds. A right is something that an entity has in itself. A rock on my property may be subject to legal protections, but not because it has rights. Those protections are derived from my rights. Similarly, a patch of land may be subject to legal protections on the grounds that preserving it for one reason or another is important to humanity, but those protections would again be derived from (and subservient to) the rights of humanity (considered either as individuals or as a whole). The relevant concept of a right does not allow for it to be derivative or subservient.
The other questions are trickier. When you ask if animal rights are inherent, it depends entirely on what you mean by “inherent.” If you just mean “not subservient” or “not derivative” (i.e., something that an entity has in itself), then the previous paragraph answers that question. But if you mean something more robust (e.g., do animals have natural rights that obtain even in the absence of entities capable of recognizing those rights), then we have to go down the chain of moral reasoning to reach an answer.
Similarly, “does what is right change over time?” is a question about right and wrong in a much broader sense and not a question about either legal or moral rights. Since it is trivially true that what we consider or recognize as right changes over time, presumably the question is asking us the more difficult question of whether or not there are independent, eternal, and objective moral truths that exist for all time. But this, of course, could be the subject of a hundred dissertations without being settled (as evidenced by the fact that hundreds of dissertations have in fact been written on the subject, and it is as yet unsettled). Much the same could be said about the question “where does what is right begin and end?”
I have opinions on these questions, and I could offer you my reasons for those opinions, but there is certainly no way we could settle them to the satisfaction of all the jellies on this thread in the time and format available to us.