The way I understand it:

Only when it is in range of the starbase and it only affects the income that the freighter brings in when it is in that particular sector. (freighter income is greater when it is further away) It also increases the income from incoming freighters but I don't know how the multiplier works. (Is it still higher the further away or is it reversed since it is their freighter and not yours) If anybody knows that I would like to know myself.

I think but I am not certain that since they have changed the zones of starbase influence from sectors to a radius of influence it may be possible to get the benefit in more than one sector if you build near the grid line.

In GC1 I would build them all along the route but in this game the logistics limit that.

Bottom line the increase of trade income from a home planet starbase would be better (especially if the planet is doing a lot of trade) but not neccessarily huge. However, the production bonuses from factory modules are very helpful and make the econ starbase an overall good deal.

No, it does not have to be on a trade route. The way it works is like this:

Economic Starbase "A" has enough modules to provide +20% bonus to trade routes.

Planets "B", "C" and "D" are in the "effect radius" of Starbase "A".

All trade routes to planets "B", "C" and "D" get a 20% bonus.

I have built an economic starbase next to 2 trade routes. How do i check what is my profit (if any)?

Thank you! Looks like they might be worth building after all. I'll give it a try and use Demealiz's trick to see how well they do for me.

However, I also do not know if that mini-freighter itself needs to be in the influence. I suspect though that not

In GalCiv I they did so I'm assuming it's the same until someone says otherwise. This would be another area where better documentation would go a long way.