You say you have recently bought the car - maybe the previous owner was, er, a little more gentle on the power and you are burning off some excess build up on the outside of the exhaust with it getting that much hotter.
Also, you mention the extra heat in the cabin - footwell area I guess and again this would be the exhaust that runs close to the floor. You will also notice when the car door opens extra heat coming from the use of the air-conditioning - all quite normal.
If it were the clutch you should be able to distinquish that smell from other 'burning'.
But there is always the chance something has got caught on the exhaust (not unknown e.g. plastic bags), so check it out for peace of mind.