Allergy or no? The most common reason for sneezing after a run is pollen allong the trail. Increased frequency and depth of breathing leads to heavy deposition of pollen on mucous membranes. But also wind, cold, sunlight, dust particles all stir up non-allergic or vasomotor rhinitis. Learn your cause by being tested by a competent allergist. Chicagoland has many. Check out aaaai.org.
Several. If you are allergic and have significant nasal obstruction most of the time, the decongesting effect from exercise may well allow your nose to run and sneeze after allergen exposure. Another possibility is the increased exposure to outdoor allergens when running. You often see more rain drops on the car's windshield when you drive faster the same occurs with running vs walking.