If a person had a cancerouse mole and decided to take a knife or razor and cut it off, would that work?

Absolutely not! : If it is a cancer ( biopsy in required to determine type , depth and spread ) the only way to accurately determine if it is all removed is by pathology to make sure the margins are clear! depending on cell type ( squamous , basil, or melanoma ) treatment and follow up can be different. Don't be a fool and treat yourself by playing doctor.