How many cities do I need for each victory condition? Do I only need a lot of cities if I am going for a military victory?
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.
Victory by Score:
As many as you can (as your cities population gives you points), I'm in doubt that you can win with 1 city.
I could be wrong, but the more - the better, you will have more diplomatic points, but with one city you could win also.
The more - the better (as your cities population gives you research points), but with one city you could win also (if you keep your opponent in the 'ancient' state);
There are options:
1 is enough if you gain required amount of culture points, but some strategies will work better if you have 3 (probably more cities).
Hope that helps.