Not exactly. Some birds can't pass gas in any way, burping or farting, and I know for a fact mice can't either. I don't know for sure about warthogs, I assume they can, but not all animals can actually fart.
Yes and No. Most herbivores pass more gas than carnivores due to their diets but this is usually done when they poo. It isn't "farting" per say but it's their way of getting rid of gas .
I knew my Zoo Biology degree would come in handy at some point .