Well it is all related. Taxation is built on population. PQ and farming both have an effect on max pop. Farming allows a planet to reach the cap as in your pop cannot exceed the mt/wk of food you produce. Example 8 mt/wk of food = max pop of 8 billion. So you want to have more than 8 billion? Will have to add or upgrade farming. Another thing to consider, happiness. 30% or less and your not going to experience growth on your planet.

That is something I would like to know myself. I've not seen it in the manual or in the game. I agree with you, leaves you wondering if there's some untapped potential left. Whats the difference between a PQ 7 and a PQ 8 max pop wise ?

BTW, what settings you playing with to get PQ 3? Lowest I've seen was 5 on random spawns and some 4s as a second planet in AI's homeworld

Well, here are the settings:
- Scenario: normal;
- Galaxy size: gigantic;
- Habitable planets, Number of planets, Number of stars: abundant;
- Star density: tight clusters;
- Anomalies: abundant;
- Technology rate: normal

I picked the Altarian and twicked them so to have:
- Economy: + 30;
- Morale + 20;
- Population growth: +20;
- Logistics: +5;
- Race techs: Hyper drive, Xeno Communications, Galactic Warfare, Xeno Engineering, Stellar Cartogrphy and Xeno Research;
- Party: Industrialist.

All in all, I think it doesn't depend on the settings, but on the race. I've started many games with various settings, but each time I've choosen the Altarian (bare natural or tweaked) I obtained a class 3 planet in the "homeworld" star, named "Wisp". And now that I think, in fact it is the only class 3 planet I have.
This brings in another interesting question. Since "Wisp" capped at 2.30 billions before any soil improvement, just like the class 5 planets not soil improved do, can one assume that 2.30 billions is the maximum population for PQ 5?
Methinks it would be interesting to complete the research tree (save from "Technology victory"), to update all that is updatable and to control the maximum population for each PQ (the difficulty is to reach such an advanced state without winning).