Can I be a veterinarian if allergic to rabbits?

Yes. You can choose an area of veterinary medicine where you do not take care of rabbits or other rodents. Or you can undergo allergy shots to help reduce your allergy problems. See a board certified allergist for further advice.
Sure. I see several vets who are treated successfully with allergy shots and practice very well. But might depend on "how allergic you are. " you should see an allergist. Good luck.