Civilian structures are things like resource extractors, shipyards, labs, etc - everything that's not a planetary defense thing

Killing ships is also pretty straight forward, even construction ships count towards it, as far as I know (though I'm not sure if trade/refinery ships do).

The mission screen will update with what you have left to kill, so if your original mission was to kill 15 ships and you killed 5, it'll show you have 10 left. When the mission completes there's not a huge notice, the AI says he's happy with you, and the mission screen blanks out - if there was a resource reward with the mission, you'll get it automatically.

You don't get an overt notice...it's easy to miss when a mission has been completed.

Also, you'll never be able to do all the missions. You either won't be able to get to the target or the missions will contradict each other. Pick the guys you want to be friendly with, do their missions if you can and ignore the others.