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:
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).
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.
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.
While technically you don't need to have a lot of cities, most victory conditions will get easier if you have a sizable empire. This doesn't mean you have to build as many cities as you can fit, but it does mean that you should try to have as many or more cities as your opponents.
The major exception is Cultural. Because the cost of social policies increases based on the number of cities you have, this victory actually gets harder if you control more than 3 or 4 cities. The key here is that puppeted cities do not increase social policy cost, but they do contribute to culture points, so you want to have as many puppeted cities as possible.
Best Answer
In general, I don't think there's any "optimal" strategy, everything is very situational - depends on the terrain, the units you have, the units your enemy has, specific promotions, etc. Still, I do have some methods I can share.
General Capturing
In general, to capture a city, the majority of the damage I inflict is from ranged attacks. I do use melee units, but mainly to protect the ranged ones and to soak damage from the city.
So, I move a few melee units to range 1 of the city but just let them fortify themselves there. Then I move ranged units - preferably siege units - to range 2 of the city (or more if it's later in the game and I have units with greater range), and pummel it until it drops to 1 HP, then a single melee unit can capture it.
Using melee units to attack the city is fine if you have enough of them - it certainly speeds the process - but since melee-ing a city is often costly it often leaves them too vulnerable to bombardments from the city or from other enemy units on the enemy's turn.
This is usually uniform across all ages, except that when flight is introduced I also use planes to damage the city, and also once I get range 3 units I try to soften the city as soon as possible, even before the core of my army gets there, just to make it quicker.
There's no rule for how many units you need. Sometimes just one is enough - a heavily promoted Rocket Artillery can bring a city down very quickly, then any single melee unit, even a weak one, can capture it. In the early game, when all you have is archers and warriors, you might need a few more - but almost always, when I think how many units I need to conquer a city, the decision is more about how many units the enemy has around the city, than about the city itself.
Finally, keep in mind promotions can sometimes play a large role. Bombers with the siege promotions are amazingly good against cities. Siege units with the logistics upgrade (two attacks per turn) can also bring them down quickly.
Capturing Cities on Small Islands
I understand you have a problem with this specific scenario. A few tips:
Use ships. It's true that early on ships don't do a lot of damage, but all you need is to be able to do more than 2-3 HP per turn and it adds up. If you can't reach that number, just get more ships. Range 3+ ships are especially useful, and the +1 range promotion isn't that difficult to get, especially if you level up on barbarians first. (and yes, even Triremes can get that promotion). Note: as of the Gods and Kings expansion, there are both melee ships and ranged ships, and you can capture cities with melee ships; I recommend this for small islands. Embarked units can defend themselves to some degree against ships, but they will still take heavy damage. The advice to do the majority of your damage with ranged units still stands, though.
Get the "amphibious" promotion for your melee units. It removes the penalty when attacking across rivers and when attacking when embarked, and this can really help for capturing coastal cities. Still, remember embarked units are very vulnerable - not only to other ships but to ranged attacks as well.
Once you get flight this becomes a different deal. Bombers can devastate cities and even fighters do a little damage, just make sure you have a nearby city or enough carriers. The AI does love its AA, though, I often find myself bombarding AA guns with ships and only them use the bombers. One final tip about flight - in my experience it's better to use fighters to air sweep and clear the way to bombers, than to use them to directly damage units.
Also, regarding your general problem with embarked units getting killed by barbarians - just make sure you escort them with real ships. Even Triremes have decent enough range and sight that they can effectively guard a few embarked units. Also remember the movement penalty of entering a tile adjacent to enemy units - that means a ship can effectively block the passage for other ships even when it only takes one tile in itself.