Would copper wire get hotter faster than graphite?
YES, copper conducts heat, graphite does not...
Both are very good conductors, but heat developed in a wire is a function of its resistivity. Copper has a resistivity of 17.2 nΩm, and graphite is 35 ?Ωm. As graphite is the higher resistance it will heat up quicker when the same current is applied to the wire for the same thickness.
Yes because copper is a much better conductor of heat than graphite. EDIT - assuming you are referring to application of heat energy, not electrical energy. Graphite will conduct electricity but is a very poor conductor of thermal energy. It's a bit confusing because you mention copper wire.
