Popa, some poskim (I believe R’ Ovadia Yosef is among them) permit spaying and neutering, though you’re correct that it’s generally prohibited.
I think the basic assumption behind Curiosity’s posts — that it is somehow wrong and against the Torah to promote gemilus chasadim against animals — is wrong. We may or may not be actually required to go out of our way to be kind to animals and help relieve them from suffering, but I think there’s no question that it’s a praiseworthy act. I don’t think you can find a single major rabbinical statement saying that it is wrong to rescue abused or neglected dogs and cats and take them in as pets.
On the contrary, the rabbis have always acknowledged that compassion for animals is an admirable trait. There are many stories of great tzaddikim assisting animals in various ways.
As the Ramak said in Tomer Devorah, one’s “mercy should extend to all creatures, neither destroying nor despising any of them. For the Supernal Wisdom is extended to all created things- minerals, plants, animals and humans. This is the reason why we were warned against despising food. In this way man’s pity should be extended to all the works of the Blessed One just as the Supernal Wisdom despises no created thing for they are all created from that source…”
Of course, it would be absurd to spend all of one’s time helping animals and neglect to help any people, as if only animals were worth helping. But there’s nothing wrong with specialized organizations dealing with particular issues like animal welfare. Your assumption that this has something to do with animal rights is incorrect — a concern for animal welfare is different than animal rights.
Actually, Orthodox Jews in Israel, among them a well-regarded rabbi who was responsible for a lot of Israel’s privatization, recently started an organization dedicated to promoting compassion for animals in Orthodox communities. I think this is a great thing, despite all the other important things people should be working on, because whenever there’s a neglected mitzvah or area of concern, people need to be paying attention to it and working on it. We are supposed to follow the whole Torah, not just the parts that happen to be most fashionable to focus on in any particular generation.