Experiencing itchy nose and eyes, a stye that is healing now and a clear watery liquid dripping from nose. what could this be? i just moved to a colder climate in norway but the nose dripping scared me. might this be csf leak? no head trauma.

It very likely is: simple rhinorrhea: runny nose from allergies, a cold, exposure to cold temperatures such as you suggest. The fact that you also have itchy nose and eyes suggests allergy/hay fever as a cause which would not be unusual when you find yourself in a new environment with new potential antigens/allergens. Consider an OTC antihistamine or begin a steroid nasal spray (Flonase). I wouldn't worry about CSF.