Ingested baby oil can lead to diarrhea and will not help the dry skin. Allergies (flea, food, and environmental), ringworm, and mites can cause dry skin and itchiness. Have your dog examined by a veterinarian to diagnose and treat your dog's condition.
I noticed that your dog is white/blonde, I have a jack Russell terrier and her vet told me that dogs with light fur usually gets allergies. Her symptoms were dry skin constant scratching. A cortisone shot from the vet worked miracles on her itchy dry skin and her shedding. Hope this helped .
Well, you can, but there is no reason for it, it will do nothing but attract dirt, and you will have to bathe him all the time. If you are terribly concerned about a health issue, you can at the very least call your vet to find out what you can do. Hope this helps.