So when I have two planets in a line, then a wormhole, then another planet, can I get a trade route there or will this not work because the line is broken?
As mentioned, you need to put up a starbase on that wormhole, however in your case I wouldn't do it. Instead, capture those two asteroids and put trade ports up there. This will give you an even better trade route without having to build a starbase.
Three things come to mind when I see that screenshot. First, you need to purchase a second population upgrade for that ice planet. It still is suffering underdevelopment tax (the first upgrade only reduces the tax, it doesn't eliminate it completely). Secondly, you should prioritize acquiring those asteroids. You don't even need to send your capital ship; four or five frigates can clear out the militia while your capital ship deals with bigger targets.
Thirdly, I wouldn't recommend the Rapture as your first capital ship. The Progenitor and Halcyon are definitely Advent's strongest choices for openers.
I managed to to grab the vulcano planet next to his capitol planet. I defended it with a starbase.
Good work! That's practically a death sentence to the enemy if you can keep up the pressure!
For some reason my ships won't destroy them yet I hovered next to them - and I had a colony vessel there as well - and suddenly I captured one of the extractors.
In uncolonizable gravity wells, you cannot "build" extractors. Instead you capture them from the enemy using your colony frigates (Vasari get to use their scouts instead, which is a big advantage for them in this regard). This makes them much easier to acquire but much more difficult to hold on to.
What is the deciding factor in capturing these neutral extactors? I can't seem to find out how to grab them.
Colony frigates have a special ability (40 antimatter) to capture them. It costs 60 antimatter for Vasari scouts to do this, but Vasari scouts are WAY cheaper and faster to begin with. Basically, you want to capture and hold these at all costs, since even a couple can greatly boost your enemy's income.