Question

Does the thickness of the wire affect the power of the electromagnet

Answer 1

The strength of a an electromagnetic is a function of the electric current passing through the electromagnet. Current is directly related to wire diameter (thickness). As diameter goes up, the amount of current that can pass through the wire for given source goes up.

Consider two electromagnets, each powered by identical batteries. Assuming everything else is the same, the strength of the electromagnet with the larger diameter wiring will be greater than the electromagnetic with the smaller diameter wiring.