First of all, I am very sorry for your loss. :/
I don't know of any rules regarding pets, as monks generally didn't have pets.
Buddha only gave advice to laypersons--laypersons are not subject to the sangha rules. (Buddha's advice to laypersons consisted of what certain actions would lead to, and urged the development of skillful actions and the abandonment of unskillful actions.)
Regarding killing--most of the suttas talk about being devoted to killing,
without showing mercy to living beings.
Showing mercy is where it is at. You would have to have the discernment that putting your pet down was in fact showing mercy. (Personally, I have no doubt that you did the most merciful thing regarding your beloved pet, and that it caused you pain. You were not callous with regards to your pet, and employed your own wisdom in the matter, as it should be.)
Now regarding gods: Buddhism
does recognize sentient beings other than humans. (Devas, Asuras, Yakkis, Brahmas, Pretas, etc.) All of these beings arise and pass away, just like humans do. Some have longer lifespans, some have shorter lifespans. Buddhism also posits that each being is the owner of
their own karma. No one can purify another. There is no use pleading to a god for this. You have to do it for oneself.
Dhammapada:
160. One truly is the protector of oneself; who else could the protector be? With oneself fully controlled, one gains a mastery that is hard to gain.
<...>
165. By oneself is evil done; by oneself is one defiled. By oneself is evil left undone; by oneself is one made pure. Purity and impurity depend on oneself; no one can purify another.
Just because Buddhism recognizes sentient beings other than humans does not necessarily make it theistic. You can't plead to a god to purify you--you have to do it yourself. (You may get guidance from others on how to go about it, but in the end you still have to make the changes yourself. Believing otherwise just distracts you from doing the work yourself.)