A betta will live for a while without a heater, but it won't be healthy. You're much more likely to get diseases, or death. For a 10g tank, you can pick up a pre-set heater that keeps the water about 80 degrees farenheit for $15 (here in the US). Don't keep males and females together, or males together. In a 10 gallon, you could divide the tank with a piece of plastic needlepoint canvas (or a commercial divider) that allows the water to pass through, and have 3 males living comfortably. You could have females, but that is a lot harder. I would suggest getting one male to start with, then you can divide the tank later on down the road if you see another you really like.
OR - you could have a few guppies in the tank. They are pretty and have various patterns like bettas, you can keep more of them, and you can keep males and females together and have babies (they will eat the babies though).
OR - you could get some white cloud minnows, which can live in a room temperature tank (they are temperate water fish like goldfish).
I don't know a whole lot about guppies or white clouds, so I can't give you numbers or parameters off the top of my head, but maybe someone else will chime in.
