These are some interesting questions.
I'll start with the third one:
Can an abstract construct such as a species even have rights of itself, like a right to survive (as a species)?
Here you're importing a pretty strong assertion in the "abstract construct" language. There's a lot of different ways of parsing what a species is. On the one hand, it's a type. Classically, it was a Form or Essence. We'll skip Plato here, but for Aristotle's idea of an essence. What happens is that we are recognizing something in giraffes when we look at them that makes the giraffes. To simplify that, it's not that we are making up "giraffe", Aristotle thinks it's an empirical observation.
Not everyone agrees, but the modern biological classifications building on Linneaus took species to be real kinds out there in the world. As we've refined that, we've realized he was wrong on some. And we've gotten some blurry realities where two species can intermate or even one's where they have to.
But bringing us back to ethics, I'm going to make a claim: only abstractions have rights at all, because rights are abstraction. Again, what matters is if they are mere abstractions. Abstracting is kind of what humans do and need to do in order to think.
But this question is centrally important to your other questions because it provides the scope of what we are answering. So now we can return to your middle question:
How can we defend the concept of species in animal rights? For example: What exactly is the moral difference of killing 2 giraffes out of a total population of 1 million versus killing the last 2 giraffes on earth?
On a spectrum, if we take species to be merely abstract constructs with no substantive reality behind them, then it's going to be hard to explain why any sort of action towards things based on this arbitrary distinction would matter. On the other hand, if we take them to be abstract constructs with some basis in reality, we could begin to add diversity concerns. If we take them to be real kinds (like Aristotle), we might be able to strengthen that further.
At a minimum to make it matter it cannot be merely arbitrary (or rather if merely arbitrary species matter, then we're just writing pure fiction here and we could make our ethical theory center on dice and wood grain directions). Setting that aside, there are several different but related motives I can imagine using:
- Valuation of a specific species [e.g., humans]
- Valuation of a specific kind of species [i.e., ones that seem to demonstrate some rationality: e.g., humans, dolphins, monkeys, bears ?]
- Valuation of diversity of species
- Valuation of biodiversity with species seen as integral to that
- Valuation of the planet and a belief biodiversity is important such that we need differing species.
Now the why of the valuation could be anything. It could be a utilitarian calculation. It could be Singer's pain minimization view. It could be religious. It could be spiritual but not religious. It could be Kantian. It could be intuitionalist. It could be purely economic.
Depending on the reasoning, it might be acceptable to kill of the last two hawks to save millions of momonga. Say for instance, if your preference is the cuteness of the animal or its relative intelligence. Alternately, you might consider some species to have prices and others to have worth but not price. (I for instance place the price of mold at about 200 yen).
This moves us to your first question:
How can we defend the killing of individuals of a species A in order to protect the survival of species B (in the thought experiment above)?
It would depend on our reasons for valuing specific species. To give an example of where it might be permissible, consider Peter Singer's view that what we want to do is minimize pain and suffering and that animal pain and suffering are not that different from our own (i.e., they count). If that's the reason we should kill, then it seems we would just need to calculate the net suffering produced by keeping invasive species A around.
Again, for some reason to value animals, it won't matter. If for instance, the invasive species are humans and the existing species are dolphins and you think both are rational, then it's unacceptable to wipe one out to help the other.