1) The growth is tied to your approval and hospital/medical center bonuses as well as any planetary special traits. I am not sure about the level of remaining pop to cap giving you a growth bonus. I think it more applies to the approval for population. The closer you are to the pop cap, the more negatives for a crowded populace.
2) Production is mostly based on your pop, then modifiers as peregrine23 said. The same applies to research.
You can use the govern button on the lower left to move the slider around. So the higher up on wealth you go, the higher the approval, but lower manufacturing and science. The same applies in reverse or you can focus on two at the lose of the third. So a specialized planet you will want to slide it more toward that specialty to take advantage of a 50% bonus in manufacturing, etc.
You can also affect manufacturing for planets that have ships in the govern screen. I do this sometimes when I want the planet to focus on upgrades, etc and build say a colony or constructor 40 odd turns down the road. That way I make progress toward the ship, but get extra to increase the facilities bonuses of the planet, which feed back into more manufacturing through more population and/or higher level factories.
3) As to invasions, they are a bit obtuse right now. When you are done with the invasion, remaining troops are treated like regular colonists. I believe the advantage is based on tech levels, soldiering research bonuses, faction bonuses and probably a bonus for being troops instead of civilians. Perhaps invasion fleets are assumed to have support craft for the invasion. They haven't fleshed out many an info box, so it is hard to tell for sure. I don't like to mess around, so I invade with 5-10b depending on planet size so I can get it productive as fast as possible after taking it.
