I have poured footings into water and it works fine. The concrete should be a stiff mix, i.e. the least amount of water possible. It basically displaces the water (pushes it out as it replaces it in the hole) if you put it in all at once. I usually rent a heavy duty submersible pump at a local rental store anyway to remove most of the water first.
As for the depth, it depends. Where in the country are you? If this water/ground all freezes and your footing is too shallow then it could all heave up. Where I live you need to be down at least 4 feet to be below the frost line. If you don't go below the frost line you might be better off using compacted gravel or something so it might not heave as much.