Questions about farms

They only effect the max population of the planet, but not the growth rate.

Actually, there are hard population limits based on planet quality. The wiki site doesn't get updated very often, so I don't know exactly how accurate its data is, but there is a page about population limits here: Population.

Farms will raise your population limits up to the level each planet will support. They will appear to raise your limit further, but once your population hits the planet's maximum, growth will stop.

Farms have no affect on population growth rate.

As a planet's population increases, however, its morale will go down, so you will need to build entertainment buildings to compensate. Don't let a planet go over 19 bil. population unless you are prepared to build about four entertainment buildings; once you hit 25 bil., you will need at least eight!