A skin tag is a whole different growth from a mole, and, yes, moles can
grow deep into the skin. That's why doctors take so much flesh when they
remove one.
You're right about it being painless. I had one taken off my lower
eyelid, inside the lash line, and it really didn't hurt. I was
pleasantly surprised. The needles they use are about the size of a
mosquito sucker-upper.