Is it possible to have seasonal asthma?

Yes. Tree pollen in the spring, grass in the summer, weeds in the fall, mold when it rains, and flu during epidemics, are all seasonal triggers.
Yes. This is very common. Allergies are the primary trigger for this type of asthma. In the spring it's the tree pollen; summer is the grass pollen; fall is the ragweed and outdoor mold spores.