I heard the two drugs my doctor may prescribe for my eczema cause cancer. Is this true?

Yes. If I am thinking of the same drugs (elidel, protopic), there are a couple of drugs which carry a warning that they can increase the risk for cancer. Here is the fda article on the topic: http://www.Fda.Gov/drugs/drugsafety/postmarketdrugsafetyinformationforpatientsandproviders/ucm153941.Htm.
Possibly. It depends on which drugs you're referring to. Since eczema is usually caused by allergies, i suggest that you treat the cause and undergo allergy therapy. The best one that i've worked with is lda (low dose antigen therapy). It treats allergies due to foods, inhalants, and chemicals.