What is the process of having a cancerous mole checked / removed? Do doctors usually just remove it?

Yes. If you have a lot of moles it's a good idea to have a full body skin exam performed once a year or so. If you have a family history of skin cancer or personal history of skin cancer then you should be seen twice a year. If a mole looks suspicious then the doctor will usually remove it and send it to a pathologist for diagnosis. When in doubt, take it out.