1. Is there an option in diplomacy to demand that an empire stop its war against you? (There only seem to be options for demanding an end to specific sorts of attacks, not the war in general.)
I don't know.
2. How does armour show in statistics? I add it to ships but I have no sense of its representation (i.e. another stat line in its description, for example) or how it works (i.e. do 10 armour points stop 10 points of damage from each weapon that hits it in a turn before being destroyed, or is it gone with the first 10 points of incoming fire it takes?).
It doesn't show as a separate statistic. Armor takes damage just like any other component, except that it is hit first. If it has 40 hit points, it is destroyed after 40 points of damage hit it. Crystalline Armor does what you were thinking. It channels the first X points of damage from each shot into the shields, thereby reducing the damage each shot does.
3. While I'm upgrading extraction facilities, do the ones currently upgrading still produce minerals, radioactives, etc., or are they offline?
They still produce resources. In fact, you could but the build queue on hold, and they would still produce.
4. I did a massive upgrade of my mineral miner plants from I to III (500 per turn to something like 1000 per turn). There appeared to be NO CHANGE in my production even though there should have been a massive one as far as I know; I kept checking stats under empire and it seemed to be stuck around 139k. What's going on?
Level I makes 800. Level III makes 1000. So, it wasn't as massive as you thought. Did you wait a turn after the upgrade was completed? There is generally a turn delay in the updating of some of the windows.
5. When ships are in formation how is their individual positioning in the formation being determined? (I want to be able to make sure certain ships are nearer the front of the formation, for example.)
It is something like the order that the ships were built. You can choose the Leader, but thats it.
6. Can boarders not attempt to board an enemy ship until the enemy shields are completely down?
You cannot board a ship until it's shields are at 0.
7. Under strategies, I don't understand the "damage" option where it says "damage percent per ship." Is this referring to how much damage my ships will take before attempting to flee?
No. I believe that what you are refering to is the setting that determines how much damage your ship will do until it changes to a different target ship.
8. Do tractor beams actually work? I had three each on some destroyers and took them up against some enemy dreadnoughts. Didn't appear to do anything.
Yes, they work. I don't think that there is a size limit to what they can affect, but I am not positive.
9. Do tachyon projection cannons actually work? I had TP IIIs on a veritable horde of small ships and took them up against said dreadnoughts but it appeared that the latter really didn't lose any weapons. I thought it would be the great equalizer given my small ships vs. the AI's heavily gunned ships. Are the cannons simply not worth it.
They, like all special damage weapons, have to completely destroy the target component in one shot, or else they do not do anything. Their damage does not build up or stack. Try using larger mounts.