It's too bad that most red dwarf stars are so low in carbon that extracting the carbon from the star itself to build the sphere may not be feasible. If it were feasible, it would answer the common Dyson Sphere engineering question of "where do you get the mass to build it?".
Another interesting thing is that if you build a Dyson sphere with the intention of using it as a heat engine between the star's surface temperature and the cold of deep space, the Carnot efficiency is so high that the Dyson Sphere's blackbody emission temperature is significantly lower than the star's normal surface temperature. As long as you have an energy sink (such as performing nuclear or chemical reactions that store energy, and leaving it stored), the outer surface of the Dyson Sphere could be a relatively comfortable temperature.
As the universe cools, the noise of the background radiation will diminish. Dyson posits that this would allow us to build an analog computer which uses less and less energy as time goes on-- it could run forever on a finite store of energy.
So even if the universe "ends", we might be able to build a computer which doesn't.
[1] https://www.amazon.com/Last-Three-Minutes-Conjectures-Ultima...
You probably wouldn't want to build one around our Sun, since it will eventually expand, and cook anything inside the sphere. Building the sphere would likely take millennia, and there is no reason to invest in building around a start that will eat it. Instead you would build it around a red or white dwarf. These are stable for billions or trillions of years. Meaning you could build a dyson sphere, and live on it until the end of the universe as we know it (Stelliferous era).
Red dwarfs are much less massive than our star, so you could build a smaller sphere (still relatively huge), and material property requirements would be less. White dwarfs are a bit more massive on average, but will last much longer.
Dyson spheres, bubbles, rings, whatever would not experience a day/night cycle. So would either have to build it out of a material that passively radiates excess energy out the other side, while maintaining habitable temperature. Or you build a second set of smaller rings or disks that orbit closer to the star at a different rate. The inner set casts shadows on the outer set, simulating a day/night cycle.
Successfully building a dyson sphere would likely only be done to either power a FTL gate of some sort or a giant computer. If you built a computer to simulate a universe, and properly managed the inner star's output and lifecycle, you could feasibly escape the end of the universe by simulating a smaller one for yourself.