Would a rotating skin design mitigate the heating problem limiting hypersonic flight?
TL;DR Probably not.

Assuming we would have a very good mechanical solution/design we would still need a way to get rid of the heat, i.e. a heat-sink. 

Imagine the panel (or what ever we will call a part of the skin) is rotated away from the flow (into the inside of the vehicle), than we would need to cool it but where would we store the heat? We cannot extend a heat-exchanger. So the inside of the vehicle would heat up indefinitely. 

Essentially we need to come up with a way to discharge entropy from the vehicle. This can be done by either a heat- or mass-flux from the vehicle into the ambiance. 

One possibility is related to your second idea: using a cooling system. This requires you to have a fluid propellant. Using the fluid propellant as a cooling fluid first you would not only cool your airframe but you would also enhance your efficiency. This also solves the heat/entropy problem because you would discharge mass (the propellant).

But given the fact that we assumed to have a matured mechanical design of the ‚skin-rotation‘ mechanism you could have multiple panels with ablative coating. So after the first panel is consumed you could (hypothetically) rotate a new one into the flow.