PLEASE do not bite him back! What a nineteenth-century idea. That only teaches him that it's OK to bite -- the same way that hitting a child "back" only teaches the child that hitting is OK because mom does it, so why shouldn't he?
He is only one, and even if you did bite him back, he is not yet capable of connecting that "discipline" (which it really isn't) with his own actions. He is not yet able to reason that "If I bite, I will get bitten and it will hurt; therefore I won't bite." So it's useless to do, even if you did decide to try it.
His age is also the reason that time outs will not work. I'm surprised anyone recommended them for a child this young. Again, he is too young to connect his behavior with your giving him the time out. If he were, say, two and a half to three, then he could make that connection, but at only one year old, he simply does not have any concept of what a time out is supposed to teach: "I did X, and Y happened. I don't like Y. If I do X again, then Y might happen again. I'd better not do X." He's not there yet.
Please go for a more age-appropriate and stage-appropriate reaction: Distract and remove!
Distract him with constant offers -- I mean CONSTANT since he is such an extreme biter -- of appropriate things to bite and chew. Those "million teeth toys" you mention are great but a bunch of them need to be in your hand (and dad's hand, and babysitter's hand, and grandparents' hands) at all times and you will need to be right on top of him at all times during this stage. Don't expect him to reach for them or show interest -- you'll probably have to shove them right into his chomping little mouth! It's a pain and difficult but it's just a time for constant supervision and securing him in safe places. If he has any kind of area where he can be corraled and is relatively happy with toys or other activities there, use that, but otherwise it's 100 percent supervision for a while so he cannot come up behind anyone as he's doing now. As for remove: Remove him from wherever he's too close to anyone else, since it seems that right now any proximity to humans or animals means a bite.
I know you're trying to keep him away from the animals but it's worrying that he particularly gets at them when they are eating. The dog may have taken it patiently so far and may be the sweetest dog in the world -- but it's still a dog, and they do not like to be disturbed when eating; if something comes up behind them as they are eating, they feel vulnerable and may instantly defend their food, even if they have not been doing so previously. It would only take one time for the dog suddenly to heed that instinct and turn on your son if your son comes up behind the dog while the dog is feeding. One incident would be enough to injure your son, perhaps even severely, and again, even the nicest dogs can turn if they are surprised and just not ready to take it. Please find a place to feed the dog that your son absolutely cannot access.
Six teeth at one year old sound like a lot to me, at least compared to my daughter, so he is in some fierce teething so I pity him!