Hi levander,
Yes … temperature regulation means that you will stay in a more “neutral” temperature range and not so hot in the summer and not so cold in the winter. Wool is great because it works in both directions which is why it’s used in the desert as well as in colder climates (although in the desert it would be used in thinner layers).
It traps air which can insulate you against cold and it wicks moisture away from the skin and stores it inside the fiber (up to 30% of its weight) and the moisture transports excess heat away from your body. Cotton is even better at wicking moisture but not as good at storing it away from the skin. Dry air is an insulator and moisture and humidity is a thermal conductor.
It would depend on what the flannel was made of but if it was cotton it would do a good job of wicking moisture away from your body so it would help until the flannel became moist. Cotton doesn’t absorb as much moisture as wool before it become damp. If it was wool flannel it would be more effective yet. It would also depend on the thickness of the flannel and what was underneath it. If the flannel was a synthetic polyester it would probably increase the temperature because synthetic polyester doesn’t absorb moisture.
Yes it will have a temperature regulating effect on any material but all the materials in the upper layers of your mattress including the foam, the mattress ticking and quilting, any mattress pad or topper, your protector, and your sheets and bedding will all have an effect on temperature regulation. There is more about the variables that can affect sleeping temperature in post #2 here and the posts it links to.
Phoenix