Can 10 month old baby have seasonal allergies? Always rubbing nose and eyes. Eyes are red...Running nose clear. ...

Yes. It is not very common, but i see a few new babies every different allergy season that have the classic symptoms you are describing. If there is family history, or baby has had any problems with wheezing/asthma and/or eczema, then even more suspicious for allergies. There is a prescription medicine approved for allergic rhinitis in this young age group, so i would talk to your pediatrician about it.
Possibly. Babies are exposed the same air, pollen, and chemicals that adults are. In fact, babies may be more sensitive to certain elements. Talk with your doctor this week to see if any medicine may be needed.