Assuming that we have a Dyson Sphere with a diameter equal to that of the average diameter of Earth's orbit and a thickness small relative to the diameter, then the inner and outer surfaces of the Dyson Sphere are each equal to the total surface area of approximately 550 million Earths. If we account for the loss of usable surface area due to the oceans, then the inner and outer surfaces each provide an area roughly equivalent to 1.8 billion Earths. Why exactly would you even have any other colonies, if you had this? This provides more surface area on the interior of the shell than is available on the sum total number of worlds in an immense map in GCII. If we then assume that the shell of the sphere is 1km thick and made of solid steel, then the total mass of the sphere is about 1.5 times that of our solar system, including the mass of the Sun and all the planets. If we assume that we are not capable of making use of stars as sources of resources, then we'd need the mass of the planets of approximately 1500 systems like our own (this is a rough order of magnitude approximation, not a calculated number, based on that Sun being about three orders of magnitude more massive than any other body in the solar system); I don't think there is that much material available outside of stars in any GCII map.

In terms of inhabitable surface area, Ringworlds produce the same issues from a gameplay perspective - if you have a Ringworld of the same diameter as Earth's orbit and which is a 1609000 km tall cylinder, you have as much surface area as 3 million Earth-like planets. Ringworlds even have an advantage over Dyson Spheres in that you can produce artificial gravity by spinning the Ringworld, a feat which cannot be performed on a Dyson Sphere (yes, you can spin a Dyson Sphere, but this will only produce artificial gravity normal to the axis of rotation, which means that anything in the higher latitudes will be pushed towards the equator of the sphere). Ringworlds probably don't need quite the amount of resources invested into their construction as Dyson Spheres do, but they are still an enormous investment (and, because of the differences in the assumptions I made for the Dyson Sphere and the Ringworld that I modeled, the Ringworld I modeled actually requires an order of magnitude more material than the Dyson Sphere does, due to its 1609000 m tall edge-walls).

If any of the playable races in GCII were capable of doing this, there would be absolutely no reason at all for those races to ever leave their home system except during the construction phase.

Artificial moons or planets, while not exactly practical either, are a more reasonable application of technology, and more likely to be feasible in a time-frame likely to impact the game.

Depends on the star. Red Dwarfs can in theory last trillions of years (the known universe has only been around for a fraction of that). A large star tends to last millions of years, while medium sized stars last billions of years. Basically larger star = shorter life span.

If you were to use a Red Dwarf as the core, you probably be making a smaller Dyson sphere. A Red Dwarf is a small star. The light and heat emitted from such a star is much less than a moderate size star that Earth orbits. You would want to be closer to increase the energy per unit of surface.

If you had the technology to build a Dyson sphere, you likely have the technology to remove mass from a star to reduce it to a Red Dwarf.

Oh, I just had a cool idea, space towns for extra living space, they would be very small but there would be services that would increase the tourism income and would generally make space travel fun. Also they would increase the trade route income if the trade route goes through the space town, and generally speaking it would hold a billion people within this space town, giving you bonus influence. 

perhaps an off world shipyard that would attract resources from local asteroid mining, they would be the only way of manufacturing huge ships because it would be impractical to launch huge behemoths out of the atmosphere. They also can be upgraded and you must protect these shipyards with ships.

Maybe this would go well with my class 0 planet mining suggesting, the more the features overlap the better.

beware ion storms and plasma storms, they tend to terrorize these colonies like a storm would with a normal city. This could be another feature that could overlap this idea.

Don't forget that you would need lots of tech to make these great structures, since it is logical to have each technology overlap each other to make other technologies easier to research.

Don't forgot about the space beaches and hot lady aliens.