I've been thinking that the natural manner for interstellar dispersion of humanity with relatively hard science is with permanently habited nomadic generation ships. Instead of building a ship, going to a suitable system and abandoning the ship to colonize a planet model which comes with enough issues that it might be impractical, the generation ship in this model is the permanent home of its habitants.
Dispersion comes instead of the fact that it is both mobile and self-sufficient, so that it can move across interstellar space without any particular need for speed. A ship like this could use low thrust engines such as ion drives or, closer to stars, light sails. Since space has relatively low aerodynamic drag, a very low thrust would allow an orbit that is an expanding spiral that gradually leaves the system. So assuming that the generation ship is "permanent home" simplifies requirements a lot.
Even more so at the destination since it really causes no hardship if colonizing the planets is difficult or even impossible. The ship will simply restock its volatiles and materials needed for repairs and people will go on living in their home just as before.
To be self sustaining the ship would need to have redundant everything and capability to manufacture every component from raw materials gathered from asteroids, comets, or other micro-gravity objects. It would also need to be a fully self-contained ecosystem. And have the capability to rebuild the ecosystem if necessary.
Obviously such a ship would also have to be capable of building duplicates of itself. Indeed any reasonable planner would require demonstrated ability to self-replicate before risking interstellar space.
Thinking about a suitable solution for such a ship, I ended with a design similar to real life. A very large "bubble" with relatively thin skin and filled with water. The bubble would have all the space propulsion and navigation systems attached to it. Maybe "tugs" pulling the bubble with long cables? You want smaller spaceships for mining anyway, so dual purpose miner-tugs might be good.
Inside the bubble would be "rings"; large structures of the titular shape, each of which is a self-contained habitat with spin gravity. The "rings" would float freely in the water with perfectly ordinary water propulsion systems for moving around. The "rings" will each be fully self-sufficient with all the life support and manufacturing needed. But they could rely on other "rings" for redundancy and the "bubble" would take care of propulsion and protection from space hazards. So while each ring would have roughly the population and manufacturing capacity of a conventional generation ship they could be simpler and smaller.
When a ship self-replicates, they build a second "bubble" attached to their current one and move some of the "rings" to the new "bubble". They then move on and build new "rings" as required by population growth until the "bubble" starts becoming full and it is time to build a new "bubble".
Like I said this is similar to how life does it, so it should work fairly well. Such redundancy through self-replication should enable enough robustness to colonize even interstellar space, and thus eventually other systems, provided interstellar space has enough objects to support such nomadic ships. Which AFAIK we do not know one way or another.
Now to the actual questions:
How large should the "rings" be in both population and, approximately, physical size. Physical size only needs to be credible enough for verisimilitude since nobody actually knows how much space the needed life support and manufacturing capacity would take. We can't really build any yet after all. I am more concerned about ideas for optimal community size here.
**The optimal minimum and maximum numbers for "rings" in single "bubble". The minimum is the number needed to be robustly self-supporting, while the maximum must obviously be at least twice as much for self-replication to happen. But you probably would want some safety margin to allow population growth even when building a new "bubble" is impractical.
The size of the "bubble". This is obviously the physical size and mass of the "bubble" which would be pretty much the mass of the entire structure. Needs to fit the maximum number of "rings" and support enough photosynthetic algae to provide oxygen for the entire population. It will be much safer if it has an entire ecosystem up to fish since such diverse systems are more robust.
How much "skin" is needed to contain a bubble of water in space? The skin would get punctured by meteorites and radiation damage so it should be self-sealing and repairable. You can assume the outer layer is cold enough to be ice if that helps. What kind of pressure is necessary to maintain?
This is asking pretty much, but I am hoping for some useful ideas.