Hello and welcome to the forum
In your new tank, please don't add all of the fish at once. You'll have an ammonia spike before you can build up your beneficial bacteria and get your cycle going.
Start off with one or two fish at the most. If you're going to live plant it, you really don't have to worry about cycling since the plants will use the ammonia but if you're not, Nutrafin Cycle is a pretty good product to help you out when starting your cycle.
Agree with the heater. It's hard to heat a one gallon tank. If you decide to keep your betta in his own tank, you can use a 25w submersible heater in a 2.5 gallon or a 50w heater in a 5-10 gallon.
As for your betta not eating, soak the pellets for 5 minutes or so in some tank water before you feed him. It's easier for them to swallow and chew and they won't sink right to the bottom.
Good luck and ask anything you need along the way. We're here to help :)