First, a 2 gallon is okay as long as there is a filter and once or twice weekly water changes with dechlorinated tap water or adjusted reverse osmosis water. I'm not too familiar with RO water and what you need to correct it (there're products out there) as I have okay well water. That said, I'd really recommend a 5 gallon for the best house and a nice amount of room for him to show off and really be a showcase. Breeders commonly use 1 gallon jars, but they also do daily water changes and often have some moss or other live plant to help soak up waste.
Second, they will be okay with no feedings on the weekends. Mine didn't and he was very healthy. Many betta enthusiasts skip a day of feeding for their digestive health as they can be prone to bloating and constipation.
Third, water changes will likely cure his problems. Whatever you do, DON'T use Melafix or Pimafix as the oils prevent him from breathing correctly (this is true for all anabantoids or "surface breathers"). Most fishy problems will be corrected with good maintenance schedules and good feedings and a good filter. I recommend sponge filters for bettas to prevent fin tearing (another way for fin rot to get in).
Oh, yeah, and as was said already, keeping a constant temp of 76-82 will also keep him healthy and happy.