4. As Loup said, luck affects only your attack rolls. It sets your minimum attack to (your luck %)*(maximum attack roll). So if you have a weapon that does 4 damage per shot, 0 luck would let it roll 0-4 damage each time it fires. 25% luck would roll 1-4 damage.
5 & 6. A short way? Not really. A not-very-short way? That I can do.
The important thing to know about influence is that all sources of it are independant. Having one planet in isolation generates just as much influence as an equivalent planet in a cluster of other planets, either yours or someone else's.
Planets generate influence as a function of population and the buildings you have built. In all cases, influence diminishes over distance. For planets, this is a linear relationship.
For influence bases, their modules DO NOT affect the influence output of planets in their AoE. The modules affect the output of the base only. Influence from a starbase decreases as a function of distance squared, so when placing them, put them in contact with the target planet(s). Their effect 8 spaces away (where the ring is) is small at best, usually negligible.
Influence is calculated per square. As in, each turn the game calculates the effective influence at each square, summing the distance-corrected influence from each source of influence from each race in the game. The borders are drawn where this sum of influence equals one, or where one race's influence is greater than another's. The race with the highest influence in each square "owns" that square.
Planets can flip when another race's influence is at least 4x the planet's owner's influence on the square the planet occupies. Once you reach that threshhold, additional influence doesn't help - a planet with 10x influence won't flip any faster than one with 4.1x. Also, the actual flipping event is a random roll. It may flip the first turn, it may take dozens of turns. I once held a planet at 80x for more than a game year before it flipped. There is evidence (not yet proven) that the loyalty ability affects the probability of a planet flipping on any given turn; to put numbers to this, if the probability of a planet flipping any given turn is 5% (a guess, I've never done the statistical analysis to derive the actual chance), a 100% loyaltly bonus would reduce this to a 2.5% chance.
There are two exceptions to this. The most important is the fixed Mind Control Center in Twilight of the Arnor. In both Dread Lords and Dark Avatar, the MCC gives a economic bonus and pretty well PREVENTS planet flipping. In TA, the MCC lost the economic bonus, but it's intended function was restored. It forces any planet at 4x to flip the first turn it can do so. The second exception is the building (whose name excapes me at the moment) that prevents the planet it is built on from flipping. Period. No matter what you do to that planet, it won't flip - including the fixed MCC.
Asteroid mines flip in much the same manner, except they flip far more easily because they don't generate their own influence. Getting to 4x is thus much easier. Note that asteroids are immune to the MCC as well.
I'm in the process of collecting data in order to put real numbers in influence equations. Hopefully in the near future I will be able to just give equations for all of this crap.
7. Yes. Obviously the high bonus line is more efficient, but there are times when you need all the bonus you can get. In DL and DA, with all techs researched, you can get 337% bonus on each base. In TA, this will vary as to the particular tech tree you are using, and what traded tech you have.
11 & 12. Manufacturing, research, and farming bonus tiles (group 1) do not work the same as influence and morale tiles (group 2). Group 1 tiles require a building that actually produces output to give any bonus. A factory will gain the bonus, the manufacturing capitol or a power plant (or any race-specific variant) will not. If it says anything about a % gain, it won't get the bonus. Same with the tech capitol and research coordinator building on a research bonus - they won't work. If it says 8 manufacturing/research/food production, it will get the bonus. If it says adds 25% man/res/food production, it won't.
Group 2 works differently, as none of their buildings give absolute production. You can't get a building that gives 5 morale, they give 5%. Due to this, buildings of this nature DO get bonuses on % basis. If your morale building gives 20% morale, it will give 40% on a bonus. This holds for the political capitol, stock markets, counterintelligence center, and anything else that gives a % bonus to that planet. Bonus tiles do NOT affect things like the Restaurant of Eternity or any of the morale trade goods. These give empire-wide bonuses and are not affected by tile bonuses.
13. The all research strategy works, but is not nearly as effective as it used to be. While it makes efficient use of your money in terms of planetary production, it also puts a pretty hard cap on how much manufacturing capability you will be able to generate. This usually results in having to rush buy things, which is even less cost-effective than a balanced production model.