Hello Mindy.....He's really too young for true allergies......You need to record an episode of the gag/cough to show your Vet......It may be simple like a reverse sneeze, but in a dog this young, you need to make sure it's not a heart defect! Depending where you purchased this dog, I would be suspicious.....Tracheal collapse comes on with age.....Normally, not this young....
Itchy can be anything from fleas to improper diet to environment to shampoos.....It takes a process of elimination to rule them out......I'd start w/possible fleas this time of year......Let us know what you find out......
I don't know anything specifically about the breed, but an allergy could certainly cause itching,, and hair loss.
Has the dog been that way from the beginning or has it happened just lately.