Not meaning to be a stickler, but the terminology used always confused the crap out of me. Now that the game uses a CONSISTENT terminology (in tooltops and in techs), lets stick to that. The tooltips over Manufacturing, Research, NetIncome, and Raw Production will be all you need.
Raw Production is the basis for planetary output. colony and civilization capitals give +5. As of v1.1, each point of pop gives a +1 as well (in 1.03, it used to be an esoteric formula which was most efficient at 10 pop, and because less efficient with more more). Have a Durantium Factory? this gives +2 for RAW production. The best boosts are found in Government Techs, which can give up to +100% in total. Approval is factored in at THIS stage as well. At its peak, it can give you another 25%. At its worst, well I dont know, I dont play with low approval However, lets assume -25%. Thats a 50% swing in RAW PRODUCTION, by keeping your peeps happy. The Raw Production is the basis for what you planet outputs.
Based on how you allocate this in the Govern Planet window, Raw Production is split into Base Manufacturing, Base Research, and Income points. This is where your manufacturing, research and income building bonuses come into play. After bonuses, it results in Manufacturing, Research, and Gross Income points. Hence, any reduction in RAW PRODUCTION can greatly reduce your research, or manufacturing (which is split between socal and military), or gross income totals.
This I think brings the "go for approval or go for all factories" argument to a clearer light. A dismal approval (say -25%) vs perfect approval (+25%) is a 50% difference in raw production. Meaning, without goverment techs to offset, you need twice as more dismally depressed peeps to be as efficient as the perfectly happy peeps. Is one factory worth changing the approval? It depends, in many cases yes, in some cases no, but now we can see the figures of how approval and other mods are affecting raw production.
Now, back to your question about population. It depends on techs and adjacent bonuses, to determine where the break point. In general yes, farms are worth it IMO, well over 12 pop. Simply because this is RAW production, that can be used anywhere, on manufacturing, research, or gross income results. But where is the break point? Its difficult to say again because of tech level and adjacency bonuses, and what other buildings you have. But lets do a simply example using late techs:
an industrial Sector on the outter ring of a Quantum Power Plant Hub will net you 110% manufacturing bonus (75% for the IS's base, and 35% for its level 7 adjacency bonus)
A lossless farm in a minor food hub will net you 6 food. 5 from the base, and lets assume a level 2 adj bonus, so +1 there as well.
6 more pop (mind you need to worry about their approval) or 110% more manufacturing. Well, without this 1 Industrial Sector building, lets say you already had 1000% in manufacturing bonuses. And a pop of 12. And lets assume no other raw production modifers, just for ease. 6 pop will increase your raw production by roughly 50%. That will therefore give you an indirect +500% in production from that 6 pop. 500% is way better than 110%, so take the pop.
Now, lets say you had 24 pop world, and it was a science world with 1 Quantum Power Plant (No industrial sectors) with a current manf bonus of 100%, but you find the higher tech buildings take too long to build (or take too many needed raw production points away from research), so you want to make a change. Like you need to upgrade from Research Academies to Discovery Spheres, and you need to upgrade like 10 of them Again, 6 pop, or 80% production (only a level 1 production). In this case, 6 more pop gives you 25% more raw production which gives you 125% indirect manufacturing bonuses ((6 raw production * 25%) * 100% manf bonuses)), while the Industrial Sector will not change your raw production, but can give you 105% more in manufacturing bonuses, totaling 205% (100% from a level 5 IS, and 105% from a level 1 QPP). 205% is much better than 25%, so take the manufacturing building.
Pop however in the previous example may still be worth it if you have only a few buildings to upgrade, because then you can re-divert that planet to full science once the buildings are built, and get a HUGE boost in research points. If you have a LOT of buildings, and will be upgrading for dozens of turns, and there will be even MORE buildings to upgrade as you unlock additional techs (not likely in this example, as I gave it will the highest tech buildings, but quite applicable early and mid game), then pop isnt worth it.
But of course each manufacturing building on a science planet could have been a research building The choice is up to you, hopefully you now have the knowledge to make the correct choice in your situation.
And finally, pop versus a factory on a huge world, dedicated to manufacturing. Trying to keep it simple, lets say 25 tiles available, average of 6 pop or 100% manufacturing from each tile. X is the number of pop buildings. Y is the number of manufacturing buildings. Now we need two
(6*X) * (Y*1.00) = total manufacturing
Well, its been ages since I had to do min/max calculations, so I got help from the internet. The solution to this equation is basically where the # of pop tiles equals the number of manufacturing tiles. This works out the same for any situation of food tech, or manufacturing tech, much to my surprise. Now this is going on AVERAGES, where food output is the same on all tiles, but it is not. So, adjacent bonuses do add nuance, but for the crux of the conversation, you want half your tiles to be pop to get max production.
Of course, thats a BIG sacrifice, from a planet, asking it to use birthing subsides for so long, to build that pop (and even much much longer for pop to build up on its own from natural growth, even with a hospital or whatever). so, more decision decisions decisions.....