Bettas are tropical fish, so they do much better in warmer water and do need a heater. The problem is that .5 gal is too small to heat reliably. Once the betta is in a bigger tank, though, it should have a heater so that the temperature is in the mid-upper 70s. I'm not sure if her thermometer is correct, though, because why is her home's ambient temperature so cold? The water shouldn't be that far below room temperature, even without a heater?
Bettas also need a cycled tank, like any fish. They don't produce a lot of waste and have a sort of lung to breath from the air at the surface of the tank, so they *can* survive for some time in a tiny tank, but it's not healthy for them. They can get ammonia burns on their gills and fins. In such a tiny tank, there's also the issue of the betta not having enough room to exercise.
Five gallons is a good size, because then you can heat and filter it pretty well. For cleaning, she can just do smaller water changes with a siphon. She should get two buckets for the siphoning -- one to dump in the waste water, and one to store the new water in so it can age for a day or so (and if she's got chlorinated water, she should still use a dechlorinator, even if she ages the water). There's no way that she could keep up with the dump-everything-out approach if she wants to keep any plants, anyway.
When I had a betta in a 5 gallon (lightly planted) vase, I usually changed about 1 gallon per day. It only took about five minutes, and I noticed a difference in the fish whenever I tried to skip the WC. That was with a HOB filter, albeit the vase was slightly underfiltered because I was trying to limit the amount of water current the betta would have to deal with.