I hear the best thing to do with him if he's biting is to throw a towel over the rat to be able to pick him up and move him to a neutral territory, like the bathtub to calm down. Make sure the towel isn't tall enough to allow him to jump out. The towel can be put over him and you can scoop him up and he can still breath and he's away from your hands if you kinda, twirl it up so it's sorta a pouch sorta thing and he's not gona get hurt.
Carefresh is a little dusty, but not as bad as some things, plus when you change bedding, they seem to sneeze a bit due to the change, but they should be fine in a few days. Also, if you see a little discharge from their eyes or nose that looks red, it's not blood- it's this mucus sorta stuff and it's normal, but it does mean there's some irritation. I usually have taken them from the cage to let them get some more fresh air in another room or something.
Also, when you try to get him use to you and such, you can put your hand in the cage to let him know you're not going to hurt him, but don't touch him. Move stuff around in the cage while he's in there. Obviously if he tries to bite you, remove your hand. Have treats with you too. Also I hear that getting them to trust you with like, yogurt or applesauce (or even some sweet baby foods) on a plastic spoon (that way he can snatch it and run to eat it) it'll help him bond with you in a way. You could try that too?
Good luck with the neuter!