Bettas like 77 or 78F I've noticed, and I highly doubt your house temp is that high or else you would be sweating at the computer right now. 5 gallons is the minimum to keep a healthy flourishing Betta. And ofcourse with a filter and heater.
A lot of people say 1gal is the minimum. Also, they don't need a filter, and most people prefer not to use one as the betta doesn't like water movement. One fish does not produce a lot of waste, therefore they do not need a filter if you do regular water changes.