Few More Quickies (admins and such)

Yes, you can only have 10 Administrators at the same time. However, you can promote Administrators to Ministers, they also give admin points, or Mobsters, thus freeing up space for more of them.

The cost for a City is 4 food and 10 Duranthium. Approval has no bearing on building Cities.

In the tooltip for improvements, just under their name, it lists their type. They receive adjacency bonus based on that, usually.

Yes, check the ship designer options for it to show over mass designs and others.

Depends on playstyle and what victory condition you are aiming for. Eyes of the Universe is good for knowing the general area of you opponents.

To be clear, the cost for cities varies upon which type of civ you are.  There are different costs (and adjacency boni) for Carbon vs Silicon (Silicon cities may be destroyed as well, while Carbon cities are permanent).  And the robots just build pop, no cities needed.

Regarding the wonders:

if I am Carbon, and have Monsontium, there is no contest.  I'm getting that food.

wonders with tech bonus, I try to place in a nice spot for the Computer Core (or any tech building, if aquatic) but it gains no adjacencies, so I place it carefully.  I like speed and vision, and the Brindle's bonus planet is awesome.

Since you are asking:

Carbon-based : cities are permanent, +25% pop cap per adjacency, costs D and P and Food
Silicon-Based:  cities can be destroyed, +0.5 raw cap per adjacency, costs D and P (no food)
Aquatic:  Cities are carbon, food costs much more research to build, but the first buildings are awesome, research buildings kick ass
Robotic:  Cities do not exist.  Build pop using D and P and social manufacturing, up to the planet class.

Okay, I discovered my issue with cities.  Its 4 food collected per civ, NOT colony.  Doh.  I finally noticed the little food icon on the top of the screen.  So I've started converting more colonies to purely agricultural, and destroying farms on economic colonies.  Meanwhile I'm fighting a war on two fronts, my empire is a gigantic C shape, which pretty much triples my defensive line, and I'm way behind in techs still.  Gah.

Glad you are getting the hang of it - agricultural worlds are great.  If you get around to Farmer citizens, they will help quite a bit on these worlds.

Don't over-worry about your tech gap.  Weapons techs are expensive, though weapons augmentations are cheaper at the start, so stacking a few small ships which boost your damage or recharge rate, along with a heavy capital, can make a huge advantage.  

Regarding fleets.  Augmentations do stack.  So with (for example) 20 logistic, you can place 2 medium and 5 small ships.  If each of those small ships boosts kinetic damage by 10%, your medium juggernauts do an extra 50% kinetic damage each.

Just make sure those boosts are fleet-wide, or they only affect the ship they are on.

How will I know if the boosts are fleet-wide?  Does the tool tip indicate so?  On the tech tree, they just say: '+10%.'  That can't stack by ship, can it?  That would make those specializations very strong.

On the ship design screen, the last tab is for fleet-wide (except for the thulium hit-point thing at the very bottom).  The tab just to the left of it gives boosts for individual ships (as well as sensors and range).  These components are unlocked with techs, the weapons are on the <weapon> augmentation tech path.  You can get speed boosts (from propulsion techs), faster repair (engineering techs), etc - all fleet-wide boost components are on that last tab.

When designing the ship, sometimes the component mouse-over is vague.  If you place it, and look at the ship stats on the right of the screen, at the very bottom - all special components are listed.  Mouse over this list to see the actual numerical advantage to the component (a bit of a pain, but at least those numbers are there).

Awesome.  Thanks.

Keep posting bdtgazo, you have good questions and good observations.