That said, none of these references says that it is sinful to beat your animal (ox or otherwise) or kill it, but one does say a righteous man cares for his own animal while another says God notes all animals.
However, God didn't care about the animals not owned by His people:
Joshua 6:21 - They devoted the city to the LORD and destroyed with the sword every living thing in it—men and women, young and old, cattle, sheep and donkeys.
No animal has a soul; however, all animals are creations of God or are allowed to exist by God, just as rats and such, which we apparently have no issue with killing, even though they are not something we eat or offer to God.
If someone chose to kill a person “humanely,” would that be any better than being killed in a different way, when both ways leave you without life you would otherwise have?
We are not responsible for the behavior of others, but we are to help them change their ways to accept Christ to then seek to glorify God.
For those interested in the references Dan offered:
Proverbs 12:10 - A righteous man cares for the needs of his animal, but the kindest acts of the wicked are cruel.
Exodus 23:5 - If you see the donkey of someone who hates you fallen down under its load, do not leave it there; be sure you help him with it.
Exodus 23:12 - Six days do your work, but on the seventh day do not work, so that your ox and your donkey may rest and the slave born in your household, and the alien as well, may be refreshed.
Det. 25:4 - Do not muzzle an ox while it is treading out the grain.
Jonah 4:11 - But Nineveh has more than a hundred and twenty thousand people who cannot tell their right hand from their left, and many cattle as well. Should I not be concerned about that great city?
Luke 12:6 - Are not five sparrows sold for two pennies[a]? Yet not one of them is forgotten by God.
Your assertion that God doesn't care about the animals not owned by his people is disproved by the passage in Jonah which I already cited. This doesn't mean that God doesn't sometimes prioritize other things higher than his care for his animal creations, as he did in the instance in Joshua which you quoted.
It's true that I have no issue with killing rats when necessary for the greater good of protecting human beings. I do want them killed as humanely as possible, however.
Your question about the killing of human beings is another non sequitur, much like your tangent about homosexuality. Murder is always wrong, regardless of the method. Killing animals is not always wrong, and nobody has argued otherwise. The issue is whether the Bible has something to say about the manner in which they are killed and the way they are treated while they are alive.
I have no idea what the point of your remark about responsibility for the behavior of others was supposed to be.