Possible. While specific allergies do not run in families, a tendency to allergies can. Since your mom has allergies, there is an increased chance that you may develop allergic reactions as well. Allergies typically develop following repeated exposures to something.
Yes, but..... Individual allergies are not inherited. You develop allergies through exposure to the allergens. Kids of allergic parents are at an increased risk for developing allergies in general, but the absolute risk of you developing penicillin allergy is not that much greater than the general population. Your mom's allergy should not preclude you from taking penicillins.