1) They are tied to gether - culture spread increases your allegiane and reduces the enemy's. However, culture spread is not capped but allegiance gain/loss is capped to +/0 .05/sec. The more culture you have over the enemy, the higher the allegiance change you can have. I don't know the exact formula, but say if you're only generating 2 culture over the enemy, he might get a -.01 allegiance change, but if you're generating 20 more he'll get the full -.05. [Someone correct me if I'm wrong on the mechanics]

2) Not sure what you mean? All the planet allegiances are automatically updated when you move the capital. They start generating more if their max has risen, and start losing if their max has dropped, this is normal.

3) Nope, only one, but star systems all count as one jump apart, so if your capital is one jump away from your star, the planet 1 jump away from any other star will be at 70%, which is pretty good.

4) Nope

The distance in number of jumps from your homeworld determines the allegiance cap of your colonies. Eventually, when the maps get large enough, maintaining an average of 80-90% just isn't going to be feasible.

When you move your homeworld, the caps of your colonies immediately adjust to compensate but the changes in your actual allegiance will happen over time.

Also speaking from experience. I think you can only benefit from one culture building. Meaning 10% max like the previous poster has said. I have tried 3 culture buildings and I still receive the same amount.

Basically allegiance is determined by distance from your capital. But you can only add 10% to it. But it's good to build at least one to spread your culture to enjoy the benefits. TEC culture allows you to regenerate antimatter faster for example. And also to colonize borders planets easier or defend easier as it takes a while for culture to recede. So you can mount a defense easier.

Hv fun.