As for your first question, we can't say. What will happen is that the water will begin to flash boil. This decreases the temperature of the water (due to the heat of vaporization). This process will continue until the water freezes and the boiling is reduced to sublimation.
The rate at which the water turns to gas is thus dependent on how fast heat enters the water from the surrounding environment. When preparing high vacuum chambers, they are often "baked," heated to high temperatures to make the trapped substances evaporate faster (typically they are trying to get tricky compounds like oils to evaporate, not just water).
As for your second question, it likewise depends on a large number of factors that do not appear in your question. However, in this case, the volume of the chamber will play a major part. As the water vaporizes, it is going to raise the pressure inside the chamber. Once the pressure reaches the vapor pressure of water at the final temperature (remember, the boiling will cool it down), it will stop.
Also note that both of these will depend on your pumping setup unless you carefully construct a device to avoid this. If you just have water in your chamber, you won't be able to pump it down to a vacuum and stop. The water will begin boiling and will hold the chamber at an elevated pressure until that process ceases. You would need a small pressure chamber holding the water inside the vacuum chamber, so that you could pump it down to a very high vacuum, and then release the water into that environment.
This effect can be very frustrating for those using high vacuum chambers. You clean surfaces fervently before sealing the chamber. If there is a fingerprint inside the chamber, it will start to outgas as you pull the vacuum. That outgassing will keep the pressure elevated until the fingerprint is totally evaporated. Typically, thanks to Murphy, this will occur at a pressure that is too high to conduct your experiment at!
