Is my daughter slowly developing a cat allergy? She had spots on trunk and now stuffed nose when home from college.

Allergy testing. If you think your daughter is developing allergy to cat then bring her in to see doctor for blood tests with her fp or an allergist to get tested . If it is true, then avoidance is the key to all her symptoms.
Animal allergy. It is common for cat and dog allergy to worsen in college students. While away from home they lose their tolerance and then experience more symptoms when they return home from school. See an allergist for evaluation and treatment.