How long does it take to cool down a drink with ice? If I have a bottle or container with X fluid ounces of water at temperature Y (say, fridge temperature or room temperature), and I put ice or ice sticks (composed of water) in it that make up Z fluid ounces, about how long should I wait before I can expect the drink to be optimally cold? The colder, the better (assuming I have ample time to drink it before it stops being cold!).
Is it coldest right when the ice is about melted, or just after a certain period of time of exposure to the ice?
I'd also be interested in whether having a vacuum sealed/insulated bottle makes any difference here, whether outside/room temperature makes any difference, and how the scenario changes if at all when placing the bottle in a freezer.
Believe it or not, I had a hard time finding a reliable answer to any of this from searching the web.
 A: You probably had a hard time finding an answer because it isn't straightforward.  There are many factors which can influence the best approach, and the times.
For example, the temperature of the surroundings and the insulation of your container will have significant effects on the temperature over time.  A highly insulated container will allow you to reach a lower minimum temperature, and cooler surroundings will do the same.  Placing the bottle in the freezer will also lower the minimum possible temperature (even if you take it out of the freezer to do this experiment, a cold bottle is better than a warm one).
The initial temperature of the drink compared to the initial temperature of the ice are also very important.
To determine when it will be the coldest, you need to compare the amount of heat transferred from the ice to the drink, and from the surroundings to the drink.  As long as the rate of heat transferred to the ice is greater than the rate of heat transferred from the surroundings to the drink, the system will be cooling.  I would expect to find the lowest temperature near the point when the ice is melting.  This is because melting ice requires the latent heat of fusion, and ice retains it's temperature during this process; so the ice can absorb heat while still remaining at the same temperature.  This phase transition represents a fairly large portion of the cooling ability of ice.
The specifics of when this will happen can vary a lot depending on your setup though.  Factors such as the geometry of the container and the geometry of the ice can be significant in determining the heat transfer between those objects and their surroundings (for example, surface area effects the rate of heat transfer, and geometry can effect convection currents).
A: 
about how long should I wait before I can expect the drink to be optimally cold?

That depends on the rate of heat transfer between the ice and the fluid.  And that depends (to a large extent) on the exposed surface area of the ice.  If you put 20g of crushed ice in a fluid vs a single 20g ice cube, the crushed ice fluid will cool down significantly faster.

Is it coldest right when the ice is about melted, or just after a certain period of time of exposure to the ice?

That's a complex question.  As the fluid cools, the heat transfer to the ice slows down, and the heat transfer from the environment speeds up.  At some temperature (based on the insulation of the container, the temperature of the environment, and the surface area of the ice), the heat transfer between each is identical.  That's the coldest the fluid will get.  A large amount of crushed ice might take the fluid to almost the freezing point quickly and stay there for a long time, then climb higher as the amount of ice is reduced.  A single cube might start cooling the drink and never get it very cold before the cube is completely melted.
Even measuring (or estimating) the surface area of a quantity of ice is difficult, and understanding how it evolves as the ice melts is even harder.  The heat transfer rate can depend on convection currents, which themselves depend on the shape of the vessel.  All told, this is a difficult thing to calculate directly.  This problem would more likely be attacked by measuring the temperature vs time of a few common configurations and interpolating between them.
