Additional thoughts: I think part of the problem, and why no consensus has emerged in the answers, is that we're trying to use both verbs (pooping) and nouns (droppings) here. If I have a question about why my pet is having trouble passing his solid wastes, I'm not going to think of "feces", "droppings", "waste", etc. On the other hand, if my question is about the output itself, I'm probably not going to be thinking of verbs like "pooping" or "eliminating". While any resolution of this should account for both cases (via synonyms), it might be worth thinking about which case we think will be more common and using that to choose the primary tag.
How about excrements?
This is an appropriate umbrella term for poop, pee, droppings and so on for all sorts of animal…leftovers.
I know introducing tag synonyms early in a beta is usually discouraged, but I can't help but think that there are just so many different ways people refer to crap (another tag!) that it may not be a bad idea to just plan on making these all synonyms as soon as there's a moderator who can do so.
Waiting for the beta moderators will also give us some time to think about this.
I think poop is a good tag for this kind of question and should make pooping and droppings its synonym. excrements is going to be a troubling tag because all user doesn't going to understand it (No offense @Baarn). Other suggestions are also sounds little wired too but in think poop is a common and famous term and will not going to arise any trouble.
elimination works for urination, defecation, excreting ammonia, etc.