You're right, bettas are equipped with a labyrinth organ that allows them to "breathe" from the surface of the water. Ideally you would keep them in a long, shallow tank of 5 gallons or so to allow them to quickly surface from anywhere in the tank if they need to make use of it (the 6.6 gallon bookshelf aquarium is highly recommended around here for bettas, that feature being one of the many reasons).
Bettas should be kept at 79-80F, but the warmer the water, the less oxygen it holds. I myself use a small air stone to agitate and add extra oxygen for my bettas... they also seem to really enjoy playing in the bubbles.

It's not mandatory though, because any good water filter you use will help keep the water oxygenated.
You can keep them with snails... in fact, it's the only thing mine have ever tolerated with them in their tank in the way of critters. Just remember that snails are messy and while they're wonderful little cleaners, they do contribute to the bioload.