That is a hard one ! One problem up front is that it is not possible to obtain "standard" conditions for such measurements i.e. preparing a 1 molar solution of dissolved graphite is a bit impossible (without making graphene oxide). The next issue is to write the (and only one) equilibrium equation related to the electrode potential.
For some of the oxidation reactions of graphite it is possible to obtain approximate standard potentials but these are normally around the water oxidation potential or higher (depending on pH and solvent) and not near the open circuit potential you would normally measure on at graphite electrode. The complication is that the water-graphite interaction is very limited (weak) compared to other possible equilibrium reactions e.g. from impurities.
For the temperature dependence of redox potentials; the Nernst equation is a reasonable guideline...
An electrode has to be a good electrical conductor so it is usually a metal. Now what this metal is made out of is dependent on whether or not it is involved in the reaction. Some reactions require an inert electrode that does not participate. An example of this would be platinum in the SHE reaction(described later). While other reactions utilize solid forms of the reactants, making them the electrodes. An example of this type of cell would be: (left side is the anode) Cu(s)|Cu(NO3)2(aq) (0.1M)||AgNO3(aq) (.01M)|Ag(s) (right side is cathode).