Can constantly touching and pushing nose with fingers make it bigger or change shape? Some people say it makes it smaller others say makes it bigger?

Health Myths. In general you cannot change your anatomy. It sounds like you have been hearing too many false myths.
Nasal shape. People with allergies tend to wipe their nose upward constantly that can cause a line across the nose called nasal crease and in lay terms doctors call is the allergy salute, so yes it can happen.