[jimbob]

It's one per sector. You can put as many as you want in the sector of course, but you will only get the resources from one of each type.

Right. Only one vessel gets to mine. But you can have multiple components working on that one vessel.
Now I don't know what happens if you put a mining satelite and a mining base in the same sector (over a planet let's say). Do you get resources from the first laid? The one with the best resource gather rate?

What if you put a remote miner over a neighbours planet, do you start to degrade his planet? What if you put a remote miner in the same sector as an allied remote miner, does only one work?