I'm extremely tired, but can't sleep because of my itchy nose. Is this an allergic reaction?

Possibly. The itchy nose is a typical symptom for allergic reaction. There could be other causes of itchy nose.
Possibly. Itchiness, stuffiness and excessive drainage are usual allergy symptoms. Itchiness by itself could be just from a local irritation. If it is allergic, taking an antihistamine at night should help.
Yes. Allergy symptoms like itchy nose, stuffy or runny nose can interfere with sleep and make you feel tired because of lack of good quality sleep.