One country can have only one precursor quest chain. Completing the chain will spawn that precursor's homeworld system (if you are the first one to complete the chain), or point to the already-spawned homeworld system (if you are not the first). Once homeworld systems are spawned, however, grabbing the final reward is an open race to all, with limitation posed by anomaly mechanics. So if situation favors, it's possible to grab rewards from all five precursor's homeworld.
At least, that's how the game intends to work. There are quite a few bugs in the chains right now.
There are always all five precursors in the galaxy, no matter galaxy size. Each precursor has an activity area covering 100 star systems, which can overlap with each other if the galaxy is small. Anomalies can only be discovered within that area, and the homeworld system will be spawn inside or near it, too.