It's been a couple years since I dove deep under the hood of this game, so I would defer to any whose knowledge is more recently gained.

I think the initial colony building, as well as all the "improvements" list their specific particulars in the colony screen.  For example, you look at a factory as if you were going to build it, etc.  Add them up, etc.

For your second Q, know that there is a difference between DA (which I play) and TA with regards to focus.  I will come back to that.  The global sliders affect all planets, of course, on a zero sum basis.  That is, it determines how the pot is divided among the three choices.  This is more complicated than it seems, because unused Military gets returned and even 1% Military will cause a planet not building a building to try to build the ship with all free Mil/social resources.  There are Wiki and old discussions on this Board giving the exact details. 

Players in DA who use All Factory typically set their sliders to 1 - 99 - 0 for the above reason.

In DA, focus lets a player shift resources on a per planet basis with no net loss.  I think it was 25% increase with the same total reduction in the other two.  In TA, I think it was nerfed a bit.  IIRC, it was a 20% gain for 25% total loss.

IIRC, a research improvement will produce its "nameplate" output only if the slider is at 100% research and the production slider is at 100%.  Thus, it takes both sliders at 100%, not just the production one.  Also, because the process uses integers, small changes may not result in equivalent small changes but no change at all until the next integer is reached.

I often tweak the sliders and focus to optimize the timing of the next research gain and other needs.  For example, if I have a couple colony ships building, I may test turning focus on and off in interations together with slider shifts to get what I think is the best trade off between the research gain attainment and the ship completion.