Yes they can get hoarse. I once had a dog who barked so much that he became hoarse. He could only squeak for a couple of days. I know he was uncomfortable, but hearing it was hilarious!
I wouldn't say "sore" because animals don't really register pain the same way we do, but they do get horse. I worked as a groomer at a kennel for years and we had a dog, Rascal, that barked his ENTIRE visit. When he was boarded for more than two days he would have the dog equivalent of laryngitis. He would still try to bark, but it was like a whisper, lol.
If you are having an issue with a pet, things like them making too much noise can cause them to get a coarse, whisper-like voice. If there was no behaviors that you can see that would have caused this, then it may be a symptom of something else wrong.
Yes. A soar throat is just irritation of the tissue of the throat. Seeing as animals have throat tissue, then they can get "soar throats".
No, animals cannot buy horses , but they can get sick.
Yes. However, horse's do not get "hoarse" they get "Coach".