Firstly, I'm glad to hear your hedgie is doing better
I saw that you said his skin is dry.. I recommend doing oatmeal baths for him. They are very moisturizing and help with dry skin. Just put some regular oatmeal in a sock or cheesecloth and put it in the bath, squeeze the sock until you get a milky water. Wash him with that and then rinse.
As for the quill loss: I don't have much experience with that but I do recommend you just keep him happy and don't try to stress him out too much. Put his cage in a quiet place and handle him gently- I assume quill loss is painful.
I don't know much about mites in bedding or anything like that. (Sorry!) As for cleaning, I think bleach is way too strong to use often. To clean Peggy's cage I use Nature's Miracle Cage Cleaner for Small Animals, or any other pet safe cleaner. Don't use any perfumes or strong scents.
About the food transitioning: If his food right now is a good one and you have enough food to last until the treatment is over, I suggest you do that. Food transitioning is stressful and can lead to quill loss sometimes, so when he's all good and healthy again you can start transitioning. To do this just add a little bit of the new food to his bowl everyday over the course of a week or two until hes completely adjusted. I also think you should upgrade his cage after
the treatment, as to avoid unnecessary stress on him. Unless his cage is too small you should wait. For more info you should post these questions to the Housing and Accessories
I wish you luck with your new hedgehog!!