if a civ can build a Dyson sphere, then it will. Whether it's built around their home star or another really depends on the max speed available. FTL/space warping may not be possible. If it is, then likely they will choose a star that has a good output around a host planet, or BUILD one a la Ring World, unless they can beam the energy elsewhere. Ideal stars would put out more energy than a red dwarf. Most main sequence stars are stable for billions of years. Choosing a red dwarf because they're stable for longer doesn't really make sense.
There will be some tradeoff between volume of the sphere vs energy captured and the light energy emitted that can be converted into energy. Maybe a tiny red-dwarf would be the best, as the sphere volume is fairly small. Maybe it isn't because the mostly red wavelengths emitted are crappy for whatever energy conversion they use. Red dwarfs are 0.08 to 0.5M.
There's a lot to consider, but lifespan measured in multi billions of years vs 1-2 billion of years doesn't make a lot of sense, unless they live for millions of years.