Hi...
I was not going to weigh-in here because the advice your getting is pretty sound, but I think you still need a few clarifications.
1. If you have your betta in a 2 gallon tank now, I would suggest you change 50% of the water each week, leaving the fish in the tank. About every 3 months, do a complete rinse-out of the gravel and wipe it down to remove algea and that should do it.
2. Try adding a live plant, like java moss to the tank, it will help reduce the algea growth, plus betta's like to 'rest' in it. Just use fishing line and tie it to a rock or something.
3. Melafix/Bettafix are the same thing, you can use it to treat 'mild' fin-rot, when its just at the beginning and not near the base of the fin. Use as directed and it will stop fin-rot.
4. Use a stonger medicine for advanced fin-rot, like Kanacyn. Be sure to use the proper ratio of medicine for your size of tank or it can poison the fish.
5. Most fin-rot is curable, so long as it does not reach the body. If it does, you need to include and anti-fungal medicine, like 'Jungle' brand anti-fungus.
6. 78F is fine, anything above 74F and below 82F is fine for a betta.
7. Don't worry about betta 'cycles' for that size of tank, just do the water changes.