i agree with 50% water changes per day minimum, especially on an unfiltered tank and definitely once you have the new filter cycling. i wouldn't heat a tank that small, unless you are able to get the smallest preset heater rated for tiny tanks. if your room temperature stays stable and isn't too cool i wouldn't worry about heating a smaller tank. i had bettas in bowls ranging from 1 to 3 gallons and didn't heat any of them and the fish were fine.
his fins may grow in paler than what they were before, and you will see a white, sort of transparent, edge to them as they grow back. betta fins are actually only a few cells thick in some places.