Population has no base non% modifers, its all derived from 0.1 pop growth per turn (forgive me, I may not have paid attention to the very high end techs for this).  So, want more than 1 pop per 10 turns, get +50% on your pop growth.  The easiest way is approval and xenobiology buildings.  The first tier xenobio building (hospital I believe) gives you +25%.  100% approval gives you +25% growth as well.  So thats +50% on 0.1, which is gratiously rounded up to 0.2 pop per turn.  It gets harded to grow faster than that, as other bonuses and a single xenobio building can keep you at +50%, even if your approval dips.  BIRTHING studies is where the magic happens.  This is an Age of War XenoBio Tech, its where you can put manufacturing points into pop growth for VERY high % gains.   Make sure to take advantage of building adjacency and terrain improvement bonuses as well, to keep you at +50% or +150%.  It sucks sometimes, even though I research the next tech in Xeno Bio, it wont help my growth on many planets, so disable the auto-upgrader (many reasons to do that, I frankly wish you could tell the game which BUILDINGs to auto upgrade and which ones not to).

I havent player GC2, but I've learned in this game its best once you have a basic planet setup, to have it focus on 1 thing, and have a backup to help tweak.  1 thing, between production, science, and money.  I usually base this on the terrain bonuses that come with the planet, the "potential" plots that can be opened with terraforming techs, and of course the base planet size.  Have a backup, because you usually dont balance yourself out perfectly.  You can easily have too much manufacturing on a planet, and be bankrupt, so you can dial your production down, and put more pop into economy, which only helps if you have atleast 1 economy building.

Approval helps pop growth, planetary resistance, influence and supposedly manufacturing, research, and economy, but I believe the tooltips are broken for that.  I'm eithe pragmatic or benevolent, so I always keep my approval over 50%, and I dont play insance galaxies so I'm not one of these malevolent peeps always complaining about the Large Empire Pool disapproval malus (you get that malus to an extent on any empire).  0% I believe is 67%, right on the border of the green and yellow approval colors.  You can get up to 25% buffs with 100% approval.  Like I said, I dont know how bad it gets, but I think its like at 33% when you start to see negatives.  As far as military goes, its important to keep the approval in mind when invading, as a planet with very low approval is much easier to invade than a planet at 100%.

Churning out troops is difficult, due to the pop mechanics.  I'm sure its why some malevolents rant about wanting to blow up planets without invading.  Lots of different strats.  I always ride the edge of bankruptcy, keeping all points on the circle in manufacturing and science, adjusting when needed.  Once you have a money planet or two made, with awesome bonuses, then those few can pay for the entire empire.

Culture flip is the "cold war" if you will of the game.  The Multipler means that their culture is X times yours.  You should also see a rebellion % of the colony (I never let them flip mine, so I havent seen it on my own, only other factions's planets).  I cant say for sure when the rebellion starts, maybe 4x?  Maybe lower.  Planetary Resistance also helps culture flip, so I would focus on that first (if in a hot war, its an excellent way to repel invasions).  And dont forget approval!  The reason its flipping is that the enemy has other planets by, each with their own influence (which grows each turn, and has been growing all game).  Its all additive.  The is especially true if this planet is far away from your other planets, as your own culture from your core planets arent really giving you any help.  You may have to ignore the planet you have been trading (it has low pop now I'm sure), and invade the other planets, which are causing so much influence in the area.