I second what was said above — you can certainly thank Hashem for and pray to Hashem for your pets in your own words. Few people know much about it nowadays, but in fact there is a ton of rabbinical literature on the importance of treating animals with compassion and such, and many beautiful stories of famous rabbis who went out of their way to help animals. Rabbi Dovid Sears wrote a fascinating book exploring this topic (among others), A Vision of Eden, a draft of which is available online for free. I’m sure other books and articles have been written as well.
Sterilizing animals has traditionally been forbidden by most poskim, though there are some exceptions. Rabbi Shmuel Wosner and Rabbi Shlomo Amar, for example, have approved letting a non-Jew sterilize your pet. Many poskim hold that non-Jews are not forbidden to sterilize pets.
If you see an abandoned animal, one good option would be to give it to an animal shelter that identifies as a no-kill shelter. Most animal shelters euthanize a large proportion of the perfectly healthy cats and dogs they have, because there aren’t enough people who want to adopt them. No-kill shelters are run by people who think that is wrong, and only euthanize them in very rare cases (if they are about to die or are extremely dangerous and aggressive). Since these places are mainly run by non-Jews I don’t think it would be a problem if they would spay or neuter them — it would seem more important to save them from suffering or death. That said, I don’t know whether the life of an average stray or feral cat is all that bad.