Civilization 5 cities and victory conditions

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.

The question you're really asking is "When does the combined culture / turn of an additional city outstrip the increase in social policy cost incurred by founding that city?"

The short answer is...

We know that adding another city increases the culture costs by approximately 30% of the base cost (that of 1 city). Therefore:

If your maximum potential culture / turn won't increase by at least 30% due to the new city, you are hurting, not helping, the time till your next social policy.

(This may be slightly hard to calculate, and if you take too long to reach your "maximum potential culture / turn" you're actually wasting turns.)

The long answer is...

It depends

To begin with, we need to make some assumptions:

1. When you found a new city, you can get its culture / turn maximized within a single turn by buying the necessary building improvements (monument, etc).

2. Ignore city-states, leader specific abilities, +culture social policies, and wonders. These all help produce culture, and will shift the "ideal city count" down, but do so inconsistently. To produce an "ideal" city count, we limit ourselves by era and improvements alone.

3. This list of social policy costs is accurate for the given parameters: medium map and normal speed.

And now, some math.

The 1st social policy costs 25 points with a single city. In the ancient era, your cities can generate 2 culture / turn due to the monument. (Remember, we're ignoring the palace for now)

This means that it will take 13 turns (Ceiling(25/2) to enact the policy, or 9 turns (ceiling (45/4) with two cities. We can continue this extrapolation -- 8 turns with 3 cities, 7 turns with 4 cities, 6 turns with 5 cities, and we finally reach diminishing returns at city 6 (also 6 turns).

For the second social policy, the ramifications of the # of cities gets magnified due to a larger starting value: One city takes 23 turns, two cities take 15 turns, three cities take 13 turns, four cities take 12, five cities take 10, and again, we run into diminishing returns cap out at at six cities (10 turns).

It is not until the 4th social policy that this trend is broken and diminishing returns end at the NINTH! city.

Remember -- this assumes that each city has a monument the minute it is founded.

Now let's say we've reached the classical age, and have temples in addition to monuments. Each city is now generating 5 culture.

The first policy takes 5 turns with a single city, 4 turns with two, and 3 with three.

What (hopefully) becomes clear is that we reached diminishing returns (4 cities as opposed to 6 cities) much faster when each individual city's contribution is higher. The more culture any one city is capable of producing, the more incentive there is to produce more cities. Even if you don't manage to build every +culture improvement immediately, you're still likely to come out ahead (as long as you're pre-diminishing returns).

So while the optimal number of cities changes due to any number of factors, you can probably safely not shoot yourself in the foot if you stay between three and six cities, with six being on the high end.