The green arrow you see grants adjacency bonus if you build a matching improvement on the tile. 

So, if you see +3 influence you want to build a consulate.

The 4 yellowish arrows arranged in a diamond grants adjacency bonus if you build adjacent to it. This is in addition to the bonus from the improvement itself. 

Basically, the effects on the tooltip are modified by the adjacency bonus. They are for the most part color coded. For a military tile you want to put a Hyperion Shrinker because you get the effect of the Improvement + .05% capacity for each level bonus. 

Influence is one of your possible Victory conditions. It is the immediate colored area of your empire. It is used to push out the area you control during the game. Personally I always build influence buildings as this keeps other empires from encroaching on my territory when they finally 'bump into' me.

Ideology choices are primarily driven by colonization but you can also get ones as Galactic events. Some tricks to help get more of your chosen ideology is to pick the building that grants 1 point per turn first and start placing that building as you can on all of your planets. This will accrue till you have enough to get a chosen trait. 

the term "domination" was used for the conquest victory in civ 5 and for a slightly different type of victory in civ 4 (where you were basically declared the winner once you had ~2/3 of the world's area and population). influence is somewhat comparable to that, but not entirey. it doesn't really matter how much population you have, just how many tiles you own. and the required % is higher. i think you need something like 85% of all tiles to win (mabe adjusted for number of players, galaxy size and/or other factors)

EDIT: and it's also a more peaceful sort of victory. in civ4 , you usually won by domination after conquering several other civs. in GC, you can do a bloodthirsty conqest based influence victory, too, but there's also influence buildings and star bases that spread your domain peacefully. that aspect works more like the "culture" in civ 4/5. you can even have whole rival planets within your influence without having to declare war on them if your influnence is stong enough (they may also flip to your faction after a while, much like the "culture flip" in civ 4)

When do I get a choice? When I start researching? When it's complete?

Ah, gotcha. The specialization does nothing more than unlock the next higher tech, which is the actual specialization.

Came across another peculiarity:

4) I've built constructors to mine resources that I find. However, it looks like I only get a single actual unit of that resource. It doesn't accrue. So, if that's correct, then I get to build one ship that uses that resource, and that's it? One mining rig for every ship that uses duranthium? And on top of that, I'm supposed to building starbases for other purposes than mining, like economy and culture?

Actually with the range upgrade youc an easily get 6 around a colony. 

At the top of your screen it will show how many of those resources you have. Use them, but use them wisely. They are best used offensively if you can. Durantium drivers, Antimatter missiles and Elerium beams are all FAR better than the counterparts you see early game. Putting a proto type antimatter missile with some defenses on it can WRECK pirates and almost solo a base if it had enough hit points.

Put that solo equipped boat in a fleet and that fleet can be unstoppable nearly until the begging of Age of Ascension. 

Also, note that the later mining technologies multiply the yield for a starbase.  So, if you research one of those (say, you get the 2nd-tier High Efficiency Mining tech), you can send a constructor to a mining starbase, and add that facility. Each resource on the map will now count as 2 (or 3 or whatever the tech indicates) as far as the little resource availability counter is concern.  Thus, with upgrades, you can build multiple units/structures for each resource on the map you own.

Also, for a beginner, I *highly* suggest you look at the Tech Tree view, rather than the Advisor view. It's clearer.

Yes.  Shields, point defense, and armor.

easily 6 but with a little planning 12 is possible