I have a VERY painful jaw which seems to be under a tooth in which I have a crown. Would it be possible to need a root canal? If so, does the dentist take crown off to do the work?

Yes and yes. It might also be possible that the tooth cannot be saved and must come out. That has happened to me - the pain was an infection from a vertical crack in the tooth, not a cavity.

Yes it is very possible and a crown does not stop other problems, often many will latter get bad and have to have more work or be lost

It is possible you need a root canal. IF you do, the root canal can be done through the crown. The DDS will drill through the crown then, after the root canal is finished, a filling material will be placed. I hope this helps you out. Good Luck!