Does removing a (non-cancerous) mole reduce the chances of skin cancer developing in that spot?

One should know if the spot is actually a mole or another pigmented lesion. For example, seborrheic keratoses develop very quickly and can be pigmented mimicking moles. I am unsure what the exact situation is in your case. I hope your dermatologist at least used a magnifying lens - a dermatoscope to check it. Removing the benign mole may not be necessary, but if there is any chance it can be atypical it should be removed. Sometimes early atypical moles are hard to diagnose, so I would advise seeking another opinion if you are worried. And you are correct, the history of tanning bed use is definitely worrisome.