I still have trouble making a direct correlation between different sets of numbers having to do with planet economy but things seem more in line than before....
What exactly do you still have trouble to understand?
As for your original question, population just provides tax income, while taxes, and other forms of income, allow you to finance your production. That's all they have to do with it.
Factories simply increase the max production capacity of your planets (same for research labs and your research production). There are other factors, that further increase it, like your bonuses to the Social and Military Production abilities (or Research ability in case of labs), planetary improvements with percentage-bonuses, economic starbases with production-assist modules in the area, etc.
The sliders in the Finance Management screen finally determine how much of your max production is used (Production Capacity slider) and how your money is distributed between each of the production categories (Military, Social, and Research sliders) in order to finance them. If the Production Capacity slider is set to 0, then you don't produce anything, period. If one of the Distribution sliders is set to 0, then you're not producing anything in that category (for example, if Research is set to 0, then you're not researching).
The sliders work for your whole empire, but you can individually fine-tune your planets a little by focusing them on a certain production category. You can do this either in the Colony Management screen, by clicking on the small button behind the production values at the top, or in the Colonies tab of the Civilization Manager, by clicking on the production value of the category you want to focus in. Focus reduces the amount of production in two of the categories by 25%, and transfers the value to the category you want to focus in (although with a 25% penalty).
For example, you're producing 40mp in social and military production and want to focus in research. This will reduce the social and military production to 30mp (-25%), and increase research by 7rp (10 - 25%). The actual value may vary, because certain bonuses are applied differently. You also have to keep in mind, that the game always rounds fractions down in those calculations (7.5->7).
I hope this explains.
