1-Research the Logistics tech
2-(answered)
3-What you see is what you get: the only ships in the fight are the ones on the combat viewer. If the starbase is on the viewer, then the starbase is fighting. However, if the starbase has abilities
other than weapons - such as the weapons assist modules - those will affect the ships in its radius, allowing it to indirectly assist in the combat even when it isn't actually fighting.
4-Mostly yes for the first, yes and no for the second.
--
Technically, labs don't actually represent research production, but rather the
capacity for turning funding into beakers. That said, the more labs on a planet, the more beakers you can fund the production of.
--For the most part, a lab is a lab is a lab, regardless of where they are. However, the Technological Capital and Research Coordinators make it a very VERY good idea to try to specialize planets for technology. One world with nine labs and a tech capital will give more beakers than three worlds that each have 3 labs, and one of which has the tech capital.
5-*shrugs* Could be. I haven't run into this yet.
6-Not that I'm aware of, unfortunately. That's why you should try to conquer in bursts, so as to pick up as many planets as possible on the turn that you push an enemy over the edge and into surrendering to somebody.
7-You get their planets, and sometimes a ship or two (depending on alignment). I don't think you get their research, though it's been too long since anybody surrendered to me for me to remember definitively
8-Generally, "the tech to the right" is better than the ones to the left. As for skipping, it depends on the tech line. Many techs give you inherent bonuses for researching them, making it worthwhile to research them even if you've traded your way past them. A couple bigger examples of this are various colonization techs, the miniaturization techs, and pretty much any "yellow" tech. If you aren't losing any bonuses or prereqs though, there's rarely much reason - if any - to go back.
9-Can't help here since I haven't noticed this happen before.
10-Governors are just a way to cut down on micromanagement. You don't have to use them, but if you want to do something on an empire-wide basis - like telling ALL your shipyards to stop building Frigates and switch to making your shiny new battleships - the governor will let you do that all at once, instead of going to every single shipyard individually and telling it to change its build orders.
11-Not that I'm aware of.
12-The easiest way is to equip more/better engines. Stick an extra couple engines on a ship, and it'll move faster. Researching better engines gives two advantages: first, it provides faster and smaller engines to put on your ships, and second, the first tech of an engine "type" (i.e. Warp Drive I) will give all your ships a +1 bonus to speed. There's a couple other ways (i.e. there's a trade good that gives your ships +1 speed), but putting more, faster engines on ships is the primary way to speed up your ships.
Now, to see if somebody came and answered everything while I was typing...

Edit: Wow, and by almost 10 minutes, too. Geez I type slow
Edit 2: I also see I mixed up "build ships faster" with "build faster ships"...