Question

How electrons produced at zinc electrode i.e anode in a dry cell move through wire and produce electricity
Do they move through wire or directly through electrolyte to cathode

Answer 1

The chemical reactions in the dry cell causes a develop of electrons at the anode. This results in a potential difference between the anode and the cathode. The electrons wants to rearrange themselves to get rid of this difference. Electrons are repel each other and try to go to a place with fewer electrons.
Electrons are moves through the wire which is connected between anode and cathode.