Could allergies and exercise induce asthma?

Yes: Allergies and exercise are 2 of the most common triggers for exacerbation of asthma symptoms. If your asthma is not well controlled and you have allergies, treatment of the allergies is an important part of asthma therapy. If exercise is making your breathing worse it doesn't mean you can't do it. It suggests that your asthma is not well controlled and this should be discussed with your doctor.

Yes: Both can induce asthma flares.