Hardwired Towel Warmer
Your #1 and #2 have a deep relationship so let's answer those together. Your GFCI has a LINE and a LOAD side. The LOAD side is protected by the GFCI. As long as your electrical device is not in an area it would directly come into contact with water (i.e. inside the shower/tub), and is attached to the LOAD side of the GFCI, it is compliant. In other words, it isn't just a good idea to hook it to the GFCI, it is required by code.

Your third part is a bit confusing. The answer stands if you're hard wiring it in, or just adding an outlet the rack can reach. As a matter of course, I would recommend wiring up an outlet rather than hard wiring the rack. If something happens to the rack, you still have a functional receptacle, rather than a strange box with capped off wires and a blank face plate.