1st of all, welcome to betta forum! :)
don't feel bad, we all have to go through learning to get to where we end up.
about food, maybe you can try other freezedried (example-daphnia, brine shrimp, krill) or even frozen foods. they have a strong smell that might entice him.
there are water conditioners that remove chlorine, which can be harsh to fish, found at the petstore.
I would get a heater, yes. you said you wanted to get a 5 gallon? the heaters go by gallon size usually. In addition I would get a thermometer so you can monitor the temperature so it doesn't become too hot or too cold. I found personally that the stick on ones are no good compared to other kinds. as said above, 75-82 degrees is a good temp to keep the heater at.
I like filters, they help keep the water purer, but just make sure the flow of water isn't too harsh to sweep him across the tank all the time.
salt (aquarium salt- not the table stuff :P) can be used in small quantities in the tank if you want.
and if you want to cycle you can, but you don't HAVE to. just make sure you change the water frequently.
as said above again, they love plants and hiding places, since your boys fins are a bit ragged, you might want to get silk plants, which are soft against their fins, and smooth places to hides, as opposed to rough of course.
good luck! and I can't wait to see your changes, I am glad you found us! :)