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.

where I live shops have bettas in and within days they are tatty or have disease!! I try not to (as it encourages shops to buy more fish!) but sometimes, no lots of times, I buy sick fish and attempt to get them better. I have about 12 fish at the moment housed in stackable plastic containers. Every alternate tank has a heater so the tanks are warm. My fish are thriving, bubble nesters and are in small unfiltered tanks. If I were a betta and looked as happy and healthy as those in the photos above I would be content!