Regarding cancer cells and telomeres
In a normal cell, during each replication the telomere is shortened slightly due to the end replication problem, as you probably know. As mutations occur and a normal cell begins to exhibit cancerous characteristics, it needs a way to stop the self-destruction which happens when the telomeres become too short. In fact it is the cancer cells themselves which 'short-circuit' the shortening process and cause the telomeres to stay long enough.  

90% of tumors do this by activating telomerase, an enzyme complex which elongates telomeres and keeps them from getting too short, so that the cell is effectively immortal from this sort of destruction. Usually telomerase is not active in most cells, except for stem, germ, and hair follicle cells, but cancer cells use this to their advantage and activate it to stay immortal. The other 10% or so employ a method called ALT, or Alternative Telomere Lengthening, a not yet thoroughly understood process which involves the recombination of tandem repeats between sister chromatids.  

This page explains how cancer cells bypass the telomere shortening process well, and also provides a series of further reading on telomeres and telomere shortening.