As far as his skin- the dandruff and unwillingness to groom can be due to several causes.
I usually see dandruff start to form when there is a lack of grooming- which is what you are describing. Not wanting to groom can stem from several different issues (oral pain due to dental disease, being overweight, general malaise secondary to underlying illness). Sometimes the dandruff can be due to a primary skin issue- like skin allergies, or nutritional deficit. To fully determine what the underlying cause is, he will need to be examined by his veterinarian. In the meantime, you could start him on an omega-3 fatty acid supplement. The one I recommend can be found here: http://www.nordicnaturals.com/en/Products/Product_Details/514/?ProdID=1592
Now, to the issue of getting him into his carrier, and making his transport and visit to his veterinarian a more positive experience.
The first is getting him used to his carrier- keep it out in the open, feed him frequently in it, give him treats in there- so that it becomes a daily part of his environment. This will take time. Once he has no fear of going into the carrier, start closing it for short periods- while providing treats to make it a positive experience. Then, once he is used to that, take him in the carrier out to the car for a few minutes- giving treats, and gradually work your way up to car rides. This requires repetition, and consistent positive reinforcement. Make sure it is a large enough carrier so that he can turn around easily without being squished. It should be a carrier that opens from the top as well as from the front, and has a top that is easily removed with quick release snaps. When in the car, the carrier should be covered with a large towel.
If there is one in your area, I would recommend a cat only veterinarian, to reduce the stresses of dogs barking. If one is not available, let the practice know how stressed he can become, prior to your visit. They may have you keep him in the car until a quiet private room is available.
The other option is to find a veterinarian in your area who makes house calls. You can search house call veterinarians online.
I hope this has helped, please let me know if you have any additional questions at this time.