I recommend not "investing" in a city-state unless you can get some influence for free to start out. Killing Barbarians or doing another quest for them will get your foot in the door, and let the $250 you have to spend periodically give you a lot more value. For instance, that $250 will be keeping you above the "Allied" line instead of just barely making you friends with the city-state.
Likewise, as mentioned above, the Patronage tree will give you significant bonuses to your influence over city-states.
It seems that the design is such that in the early game you won't be able to guarantee a city-state ally without committing a lot of cash.
To elaborate (not 100% sure on all these numbers but the overall point is still valid):
If you spend $250 to become Friends with a city-state, starting from 0 influence, you'll get 35 influence, making you barely Friends (30 influence). This means you'll be friends for 5 turns, at which point you'll need to spend another $250 to be friends again. You probably don't have another $250 that soon, though, so the influence runs all the way down close to zero before you're able to spend another $250, which again only gives you a few turns of friendship. This is obviously unsustainable.
If instead you perform a mission for the city-state and get 30 or 40 influence to start with, every time you spend $250 you'll get to use all 35 points of that influence before needing to spend another $250.
Similarly, you'd be better off saving your money to start with $500 worth of influence than buying $250 and not being able to afford more right away.
Long story short: if you're already friends, you get more value for your money. Likewise, if you're already allies, you get even MORE value for money you spend to sustain at that level.
Is "Micro-Empire" a viable strategy for one or all of the victory conditions?
Micro-Empires don't work for all victories. Almost by definition, you probably won't win a domination victory with a micro-empire. To do so you would need to raze all cities you capture (to stay micro), which would just give your opponents space to expand, which makes your job harder. It would also be difficult to balance advancing your cities and cranking out military units without falling behind on both fronts. I think a minimum of 4 - 6 "core" cities is a must for a domination victory. It may be a great challenge to try it with two though.
Having said that, in addition to @WillfulWizards hints:
- To stress the first point a little more: be Switzerland. Stay out of all conflicts, ignore all invitations to wars and pacts of secrecy. Only agree to positive deals. Listen to what the AI players are saying, they're rather straight shooters. Even if they ask you for contributions of gold with no return, if their intentions are good ("It's not going well over here, could you spare a little, I'll return the favor") it'll strengthen your relationship. If they demand, you already did something wrong (or you're up against a hostile AI) and should do your best to improve your relations.
- Even so, you probably can't avoid a war forever, so dig in. Get as many defensive wonders as possible. The Great Wall, Himeji Castle and the Kremlin are great wonders to have if it ever comes to a war. Build no roads, or only those necessary to establish trade routes,
to slow the enemy down in your territory as much as possible. Adopt defensive policies.
- Settle in a good spot near as many city-states as possible (this obviously requires some luck). If you stay allied with them they'll help fend off invading forces.
- Go for food and growth in the beginning (build farms), transition to gold/culture/science mid to late-game when your cities are huge (replace farms with trading posts/landmarks/academies).
- Hand-tweak citizen allocation, at least lock certain fields like production and set a focus for the remaining citizens.
Best Answer
Dispersing a Barbarian encampment awards no culture, only gold.
If you have the Honor policy, and kill a Barbarian unit, you get an amount of culture proportional to the strength of the unit. I believe this is 50% of their strength, although I believe it's something they've adjusted previously.
Montezuma of the Aztecs gains a similar bonus for killing all units, and these two culture bonuses stack.
The underlying question you're trying to answer is "is it worth it to farm Barbarians for culture?" The answer to this depends on the amount of Barbarians and the amount of farming you can do. In general, if you want culture, I'd suggest either Liberty or Tradition instead, since they both provide flat culture-per-turn. Barbarian units don't tend to spawn fast enough to beat the amount of culture you'd get from these other policies.
If you're going down the Honor track for other reasons, then the answer is "possibly" - if you're not at war, and have more military units than you need to defend your borders, you may as well farm them for the small culture bonus.
Do note that if a camp is near enough to a body of water that naval units will spawn, there is a cap on how many units a camp can have active at once. Thus, you'll have to destroy the naval units if you want more land units.
Also note that the AI doesn't really think about or care about farming Barbarians, so they will often march up and wipe the camp off the map, taking the gold and leaving you culture-less. :(