1) Is there a way to mass move from Hatchery to Stable?
2) Is there a way to mass move between tabs in the same stable?
3) I see something called Stable Hand... Do you have to be a certain level to unlock this feature?
4) How does the food thing work? Is it a feature that is currently frozen? I'd hate to have all these wonderful pets, then come in one day to no gold because it got unlocked.

I really appreciate any and all help. I'm sure there are answers to these questions somewhere, but I just haven't been able to find them. I'm guessing they are right in front of my face and I can't see the forest for the trees.

1) Nope
2) Nope
3) It's not implemented yet! But soon!
4)It's currently frozen, it will be reinstated with the stablehands.

Divinis answered all your questions. Just adding on for Q4.

You don't have to worry about losing gold. Each pet has a hunger bar (it's that yellow strip under genetics). So it takes a few days for them to get starved.
Even when they are starved, the gold won't be deducted from your account. You can choose to feed them or not. If you don't, then they'll fall unconscious and you won't be able to use them in the game.

I believe once the stable hands are implemented, you can hire them to feed your pets for you. Or don't hire them and feed your pets manually. But I'm not too sure about that. We'll have to wait and see once it's implemented.

Does anyone know if the revive a pet option is also frozen or just not working? I tried it and it woudn't work so..

Make sure you removed the unconscious pet from your party before trying to revive it. It should work then.

OOh ! Oke thanks I'll try that :D