The reason why your cat curls up to sleep is a combination of the obvious and something that probably hadn’t occurred to you. The obvious reason is that the cat keeps warmer when curled because the shape of the body makes it a more compact mass which helps to retain heat. The surface area is reduced because when a cat curls up the maximum amount of the body is in contact with itself. This keeps heat in the body rather than it being dissipated to the atmosphere.
A less obvious reason is depicted in the comparison collage above. Curling up is a evolutionary development for survival because the original domestic cats were all tabby cats and their ancestor, the North African wildcat, curled up as did the ocelot and other small wild cat species so that they look very much like a snake, which is feared by many other animals or at least many animals are aware of the danger to life that the snake poses.
When a cat sleeps she is vulnerable. The cat gains an automatic level of protection when looking like a snake. Cats circle when settling into a curled up position. I don’t think this is significant. It is simply the cat settling down. It is almost as if they are curling themselves up while walking/circling. It just makes it easier to be in the curled up position when settled.