In a 1 gallon a filter is pretty useless (you should be doing weekly 100% water changes) and I'm not familiar with any heater small enough... Basically, a decoration and/or a silk or real plant are your options for tank decor. My fave staple food for bettas is Hikari Betta Bio-Gold pellets. The frozen worms everyone talks about can be found at most LFSs... they come in little packages that will last you FOREVER with just one betta, and there are several suitable varieties, including blood worms, adult brine shrimp, mysis shrimp, and white mosquito larvae. Bettas go crazy for them I tell ya! You can even get live worm cultures if you like, which can be kept in a shoebox-sized plastic tub of dirt and will give you an endless supply of live food. You can also feed your betta a bite or two of frozen pea on occasion, but no more than once or twice a week since bettas are primarily carnivores and need a lot of meaty foods (such as the aforementioned frozen foods) in their diet.
As for your fish, he is definitely sick. In my experience, true fungal infections are relatively uncommon and it's probably bacterial. My bet is columnaris, in which case you should try to find a medicine called Maracyn II at your LFS with which to treat it. You can also try Melafix or Bettafix, but use a quarter of the dose of Melafix recommended on the bottle as bettas tend to be sensitive to it. Adding a pinch of
aquarium salt (not sea or table salt) to the water will also help, along with daily water changes. Change the water before you add any medicines, then medicate, and repeat the following day
