Is it possible to have "the sniffles" without having an actual cold?

Yes: Allergies are a common cause of the sniffles, but any irritant in your atmosphere can cause them.

Yes: Runny nose and sneezing occurs when there is irritation of the nasal lining also called rhinitis. This can happen from a cold (virus) or from other allergic or non-allergic causes. Aside from the common cold the most common cause of rhinitis are environmental allergies.

Allergies.: That sounds like allergies. Nasal congestion, runny nose, cough, headache, facial pains and earache can be triggered by fragrances, dust mites (sweeping the floor), grass, trees, pets, environmental molds and such. There are outdoor and indoor allergens. OTC nasal steroid and nasal saline are worth a try.