Its really easy to balance happiness with a large number of cities if you plan it right and choose the right production queues and social policies. On my current king game I have around 20 cities and a surplus happiness of around 20, and thats with only 5 luxury resources (as the new patch makes trading a tougher effort).
My favourite methods include keeping each city smallish (around 5-7) in size, building Colosseums and circuses early (after monuments, libraries and markets), making sure you get the meritocracy sp (+1

for each trade route - liberty branch), build the forbidden palace asap and preferably keep cities close to each other to lower road costs.
I am different from ICS players in that I don't recommend going the Order policy branch route, mainly because you have to adopt 2 duff policies before you get to the good stuff, AND wait all the way for the industrial era to spread.
Instead, after hitting meritocracy - buffer your culture until the renaissance then adopt the first policy of Freedom (specialists produce half

) and the first two of Rationalism (5 turn golden age, and +2 science from every specialist). Now with all but your best production cities make sure you work as many specialists as you can (in most cases 3 per city - 2 from library and 1 from market). Not only will your happiness dramatically improve but your research beakers will skyrocket and allow you to expand much quicker. Tile improvements are secondary to specialists once you unlock the Rationalism +2 science from every specialist sp in my opinion.
Also I hardly ever build military units except for early on, it takes too long. I simply buy them and keep upgrading my old units.