When installing the console over carpet, should i cut out the carpet at the mounting areas, or just mount it through the carpet?

Cut it out

yep, cut it out. It'll also help reduce gap (if any) between the console and carpet.

Would you recommend cutting out the whole carpet under the console. or just the mounting areas?

I just cut the pieces around the mounting areas. In fact, I really didnt cut any patches out, just slit the carpet so that the mounts (shifter and rear mount) stuck up through the carpet.

Console doesn't cover as much as you would think, especially on driver's side of tunnel ( if it's auto) white crayon used for marking fabric helps alot.