In what ways can mechanisms of apoptosis be damaged?
Lot's of ways. Apoptosis is complex, but falls under two pathways ending up at caspase 3. Anywhere in the pathway may there be a problem but also in things that trigger the pathway. For example in cancer there is loss of tumour suppressors which ensure a damaged cell undergoes apoptosis or prevents replication and oncogenes which allow controlled replication. Damage to these genes allows a cell to divide in the absence of signals to divide and also forget to check that it's DNA isn't damaged before replicating. This is then passed on to the daughter cancer cells. As each check is removed the cell permits more and more mutations meaning more and more likelihood for the next mutation to occur. Cancer is caused literally by one cell having defects and every daughter cell also possess the defects. Those that have defects which accidentally kill the cell are selected out.