Just use tap water. But make sure you add some sort of conditioner to the water to remove the chlorine and chloramine.
As with any tank, but
especially with a 1/2 gallon it is very important to do regularly scheduled water maintenance. With something as small as a 1/2 gal. water changes have to be done more frequently. I personally do one 25-50% water change mid-week and then a 100% water change once a week (during which I wash the gravel, plastic plant, and tank with hot tap water). Some may think a 100% water change is too stressfull for a fish...but frankly with a 1/2 gallon with no filter it's necessary for the overall health of the fish and overall water quality. When I do the 100% water change (BTW the new replacement water for all my water changes has already been conditioned in a separate jug at least 24 hours before the change to allow it to come to the same room temperature as the water the fish are already in), I add one drop of Aquarisol as an ick (and other parasite) preventative.
As for the eating...it takes some fish a while to get used to their new environment; to settle down before they start eating regularly, especially picky bettas. And I'm sure the "disgusting" water conditions weren't helping him at all. The water change should help him quite a bit. And as I've said before, I would experiment with some other types of food.
As for your heater question...do not, I repeat: do NOT use a heater with a 1/2 gallon. It's too small for a heater and you'll fry/boil your little dude. Room temperture should be just fine as long as it's in 64-82 degree farenheight range (with preference being in the mid to high 70's). Get yourself one of those little stick-on thermometers for your tank so you'll know what the temperature of the water is. My guys are kept at room temperature...and where I keep them it's normally over 75 degrees (the last couple of days it's been right around 80). If you live somewhere where the temp goes down considerably during the evening, I read someone else's post that suggested wrapping towels around the tank to warm it up. Heck, even a regular table lamp near it will warm up the water a bit for him.
Hope this helps. Also...go to your LPS, LFS or even a good bookstore and pick up a good book on bettas (the one by Mic & Maddy Hargrove's a good start).
Keep us posted on Mitch