oh dear! poor little thing, glad you got him
and it really sounds like a rescue, not like a "rescue"!
first of all, food. they have very sensitive digestive system and any change in food needs to be done very gradually by adding new food and reducing old food - over about a month or so. the food you've got from your guinea pig neighbour.. i would suggest you give him just hay rather than introducing guinea pig food. he can have some bran but i'd take the raisins away. as for vitamins & minerals, the main two things are vit c and calcium - they would normally get this from a healthy diet - i would wait for the vet visit. they will check his teeth to see if he gets enough calcium - no point in you trying if you can't put him up and there's not that much you could do until the vet visit anyway. also, he was getting bran and cereal is fortified with vitamins and minerals.
keep monitoring his poo. they are pooping machines and should constantly produce droppings. these should be smooth, with round edges. if they are small and dry - not good. if they have sharp edges - not good. if they are so soft they are easily squashed - not good. if they stick together - definitely not good. if they are liquid - panic, code red, emergency vet.
raisins can cause diarrhea and if he's been eating these a lot, taking them away could cause constipation. if you do notice his poo going hard and small after you take the raisins away, i would give him a few and wean him off them slowly.
dust bath will help his fur, it will take a while. they also shed fur quite readily so once he gets to know you better, you may be able to tug on them and they come out. my rescue gets a little clumps of fur around her mouth because i need to give her medicine and liquid food and some of it goes on her fur. they do come out although she's not too keen
i wouldn't shave or cut them but i would try to help them out if he's happy for you to do that. you could also try brushing him with a soft cat hook wire brush if he lets you.
cage, he would very much enjoy having shelves and ledges so perhaps you could look into getting him a chinchilla cage in the future but i think for now he's delighted with what he's got after living in a cat carrier.
it's very encouraging that he already lets you stroke him!
well done! good luck! xxx