I was thinking about enzyme catalysis, and it seems like enzymes can only catalyse one kind of forward/reverse reaction (please correct me if I am wrong). Does there exist an enzyme that can catalyse multiple reactions with different cofactors? For example, an oxidoreductase that has $Fe(III)$ as a cofactor for one reaction and $H_2O_2$ as a cofactor for another reaction.
It sounds like you are asking about what are commonly referred to as "multifunctional enzymes".
For a reasonably recent article covering this subject — see: Cheng, X. Y., Huang, W. J., Hu, S. C., Zhang, H. L., Wang, H., Zhang, J. X., ... & Ji, Z. L. (2012). A global characterization and identification of multifunctional enzymes. PloS one, 7(6), e38979.
A specific example of such an enzyme is the fatty-acyl-CoA synthase from yeast — this enzyme catalyzes multiple reaction steps that use both NADP and Coenzyme A and also uses a third cofactor, FMN.
The enzyme from yeasts (Ascomycota and Basidiomycota) is a multi-functional protein complex composed of two subunits. One subunit catalyses the reactions EC 18.104.22.168, 3-oxoacyl-[acyl-carrier-protein] reductase and EC 22.214.171.124, 3-oxoacyl-[acyl-carrier-protein] synthase, while the other subunit catalyses the reactions of EC 126.96.36.199, [acyl-carrier-protein] S-acetyltransferase, EC 188.8.131.52, [acyl-carrier-protein] S-malonyltransferase, EC 184.108.40.206, 3-hydroxypalmitoyl-[acyl-carrier-protein] dehydratase, EC 220.127.116.11, enoyl-[acyl-carrier-protein] reductase (NADPH, Si-specific) and EC 18.104.22.1689, (R)-3-hydroxyacid ester dehydrogenase. The enzyme differs from the animal enzyme (EC 22.214.171.124) in that the enoyl reductase domain requires FMN as a cofactor, and the ultimate product is an acyl-CoA (usually palmitoyl-CoA) instead of a free fatty acid.