A betta would absolutely LOVE a 6 gallon (12 would be heaven though!) With a 12 gallon, you could even start a betta sorority... though you do have to be careful since some get rather aggressive.

In general, bettas are pretty easy to care for. My 1-5 gallon tanks are unheated but now that it's summer, my tanks have gone from the usual 71-73 up to 79-80. The change is huge! They're much more active and non bubblenesters have started bubblenesting.

A heater for a 6 or 12 gallon wouldn't be too expensive. I paid 35 or so for a submersible one that heats up to 30 gallons I believe. My lower quality heater for the 10 gallon cost about 12 bucks.