1) Get a dechlorinator, it is healthier for the fish.
2) You do not have to wait until the tank matches the temp of your fishes cup. simply float the cup in the water and when the temps match drop him in
Swimming around means he is happy. some Bettas are lazy and swim maybe once a day. Others swim all the time! One male Betta of mine swims against his currect all day long. He struggles against it but he will get out of it and then go back to the beginning to swim through it again(like a kid and a slide. Up down up down up down).
A female of mine swims around her 5 gallon tank non-stop at times too. Yours is probable exploring his new home and is a bit anxious about the change. He could be like that forever for all you know so just le thim have his fun. Oh and get the dechlorinator!!!!