, but if i build 1 farm, my population cap will be about 10B. so 2 entertainment centers will keep the peeps happy. So for the same population would 1 later improvment, like the stadium, do the same job as 2 earlier ones?
No not for the same population. The presumption is that your population is growing the whole time. So your population may be around 6b when you have 2 ent. centers, 10b when you have multimedia, 12b (and an upgraded farm) when you have stadia, etc..
And it takes time to research to and then to build those Global Civ ones. What do you do in the mean time?
Early on you can just rely on your racial bonuses and lower your taxes. You don't have that much population anyway at start; lowering your taxes then doesn't cost that much. Xeno Entertainment and Xeno Ethics (if you choose Neutral or build Harmony Crystals) give big bumps to morale.
Building constructors to milk those yellow mining resources is good--for the price of 1 entertainment center on 3 different planets, you can instead build one constructor and bump up morale by the same amount on ALL your planets. With that kind of deal, why even think of building morale improvements until all your starbases are maxed out? And even then--research Xeno Factory Construction and build another constructor! Get another 5% global bonus.