How does having planet quality abilties effect conquered planets? If I have a plus one to planet quality and an AI who doesn't conquers it, does it become a class lower?
In general, yes. However, the effect depends on the base PQ of the planet and how high your PQ-ability is.
For example, if your PQ-ability is +10% and the PQ of the planet is 9 or lower, then there is no difference for either of you (10% of 9 is 0.9, and the game rounds this down to zero). On the other hand, if the base PQ of the planet is, for example, 20, then it would be a PQ 22 world for you, and a PQ 20 for the enemy.
Does planet quality effect any else other than the number of squares available to build on? Does a high quality planet have higher max population, growth rate or happiness?
A planet with a PQ of 11+ grants a morale bonus of +10%. Unlike other morale bonuses, however, this one isn't affected by the base approval of the planet.
Does building a farm on a planet have any effect if the planet hasn't reached it's max population yet?
No.