1) Hello, and congragulations on your new betta! He's lovely.
2) It is possible that this red spot is just part of his natural coloring. If you are sure it wasn't there before, it is possible that he scratched it. He might also have a fungus.
3) If he scratched himself, take out any objects that are sharp or rough, and put in some Bettafix, or a different fin/skin repairing medicine. Bettafix works fine on my bettas.
4) If you think it is a fungus, Rid-Ich works beautifully for me. You can also get many other remedies at Petsmart. I always have trouble finding these things at Petco, although my 3 bettas came from there.
5) The new betta is probably reluctant to eat because he is going through a crazy life changing-for-the-better situation. He might not be used to the food, so you can try a different food or give him time to get used to this.
6) It probably won't be neccessary to move him into your critter carrier.
He is blessed to have such a caring owner. And a 5 gallon tank is wonderful! Good luck, I hope he turns out fine.