Can nasal inflammation be due to allergies?

Yes. There are three main causes of inflammation in the nose: allergies, infection, and autoimmune disorders, in which your immune system is attacking your own tissues. The first two are far more common. Persistent inflammation is most likely due to allergies and is often seasonal.
Yes. Inflammation is one of the secondary effects of allergic reactions. This usually happens latter after the itchy, sneezy, drippy drainage phase is passing.
Yes. Allergic rhinitis is by definition a condition in which the internal structures of the nose become inflamed. When allergens enter the nose, histamine is released at the molecular level, triggering a cascade of responses, including edema of the turbinates, mucus production, and general inflammation of the mucosa.