Yes. Brewer's and baker's yeast are essentially the same organism. Saccharomyces cerevisiae.
Yeast. Yes, that is possible.
Sure it's allergy? Be sure that you truly have an allergy before you try to avoid yeast. Best bet is to see a board certified allergist for evaluation. If you are truly allergic to yeast, you should probably avoid breads, wine and beer that contain yeast (brewer's or baker's).
NO. You need to avoid yeast. Don't worry about avoiding wheat or gluten unless you have an allergy to that also. Unleavened breads like pita bread, tortillas, crackers and flat breads will probably be your best bet.