Home worlds have various bonuses including a big one to food. The minimal food produced is 10 megatons instead of the usual 5 for an average colony, it's basically like it already has a farm on it. Taxes and population are the major forces that cause low approval, so a tax level that is fine with your basic 5 billion tax payers colonies will generally hurt your approval on your more populated ones like your homeworld.

During the early phases of the game (after or late in the colony rush), a 50% approval is nothing to worry about on a single colony like your homeworld. It's often hard to keep up as it will have far lower approval than your new colonies and you don't have the morale techs or resources to make up for it easily. You won't get the population growth bonuses while at low approval, but more people would cause more unhappiness anyway. If approval drops as low as 40, all that happens is the population stops growing, which isn't good, but hardly anything to panic about unless you are under heavy outside influence. Once you get a big red frowny face though, it's past time to do something to make them happier. Generally try to keep your colonies happy and growing fast, but the homeworld can be an exception after you've pumped out all the colony ships you need.

No - building farms IS a good idea, since more people = increased tax revenue and more troops available . However, you need to build morale buildings (entertainment networks etc) to keep them happy, typically at a ratio of rougly 1:1 (with one for the initial colony as well) in the early game and less later on.

Hope this helps

Are you saying it's best NOT to build farms? Just keep all colonies at 5 billion mark? (Except the homeworld)

No, I was simply refering to the early game period at the end of the colonization phase. A few planets might be up to the 5 billion mark depending on galaxy settings, but in general you are going to have a much larger population on your home world and it is hard to balance with all your other small pop worlds when you have only an empire-wide tax slider. I enjoy the many benefits of keeping a larger population (tax, troops, influence), but I was just trying to help the OP out with their problem in this early portion of the game. I remember it happening to me on my first game and I thought it had something to do with the enemy influence starbases they were setting up. It made me pretty nervous watching my approval drop like that, I kept wondering why my homeworld was so hard to keep happy.

I eventually get up to at least 1-2 farms per world (except low quality planets which won't grow to that level), though I'm experimenting with 3 in my current game. Once I start getting morale boosting techs (which have innate bonuses), the more advanced morale buildings, morale trade goods / wonders, and possibly a morale resource or two, I can usually hold tax at 80% will 100% morale on most planets with only 2 farms each. Late in the game most planets will hold a single farm without even needing even one entertainment center. At 3 or more and xeno farming tech at least however, they start hitting the big morale drop offs in the 21 billion + areas and to get over that hump, it requires a lot of entertainment tiles. I've also lowered my taxes this game to about 49-59% and the main reason I'm doing this is that I'm trying to make a giant farm planet just to see how it works out. At the lower tax rate I've been able to get past the last morale drop and have it up to nearly 40 billion on that one planet with a political and economic capital, as well as the mind control center (which just gives +100% econ right now).

In general, I just try to keep my planets as uniform as possible on population so that adjusting the tax slider will create a similarly uniform effect on all of the planets at once. That way I don't have to evaluate each planet's morale status to decide how much entertainment to build, just know how many I will be needing which saves time as I'm currently trying my first gigantic / abundant / abundant / abundant game.