Is it possible to be allergic to something later on in life when you were perfectly fine before?

Yes. Food allergies to crustaceans like shrimp and to tree nuts often appear in adulthood. Part of the problem may be that these foods are eaten infrequently. If you have never had a certain pet & acquire one as an adult you could develop an allergy to it. Allergies to medications, especially antibiotics and nonsteroidal antiinflammatory agents like Aspirin & ibuprofen, often appear in adult life.
Generally not. Allergies can develop after significant changes in hormones; primarily in adolescence, but also after a pregnancy, and sometimes after menopause. But this is not common. Symptoms that develop after age 40 are often referred to as "allergies", but are typically something else. The best approach is to discuss your symptoms with someone well versed in what is allergy and what is not.