Water boils when the pressure is less than its vapour pressure (there is a table of vapour pressure vs temperature here).
At 20ºC the vapour pressure is 2339Pa, so if your balloon exerts a pressure greater than this the water won't boil. If the pressure exerted by the ballon is less than this, the water will start to boil and the steam generated will inflate the balloon. This will increase the pressure until the pressure has risen to 2339Pa, and at this point the water will stop boiling and you're left with water and steam in equilibrium.
If you heat water to 100ºC the vapour pressure rises to 1 atm (101325Pa), which is of course why water boils at 100ºC at sea level. However if you have a really strong balloon capable of exerting a pressure of greater than 1 atm the water won't boil at 100ºC even in a vacuum, and you'd need to raise the temperature above 100ºC to make it boil.
From the original assumptions:
So far, my understanding of water evaporating is the following:
The higher the temperature, the higher the vapor pressure, therefore the faster water vaporizes.
The rate of water boiling, assuming a constant boiling temperature, is dependent on the rate of heat transfer to the water, not the vapor pressure.
At sea level, water boils at 100°c. Boiling temperature decreases as athmospheric pressure decreases.
It is somewhat more correct to say that the boiling temperature decreases as ambient pressure decreases, no matter what the source of the ambient pressure.
It takes a fixed amount of heat / energy to evaporate water (once it reaches boiling point?)
The heat of vaporization of water decreases as the boiling temperature (and boiling pressure) increases, up to the critical temperature (and critical pressure) of water. This means that if pressure changes substantially during the boiling process, the heat of vaporization of the water will also change substantially during the boiling process.
To answer the question directly, there are a few industrial situations where water is given enough heat to instantly evaporate if. For the case of hot oil systems in industry, a furnace heats up the oil to 450 deg F (maybe hotter), and sends the oil to process equipment for the purpose of boiling the process in a heat exchanger. Very infrequently (approximately 5 year cycles), the hot oil system is shut down for maintenance, and water is used to hydroblast equipment in order to clean it. This means that water collects in low points in the associated piping. After maintenance is complete, the proper start-up procedure calls for slowly heating the oil, checking all low points in the piping for water, and draining all water before the hot oil gets above the boiling point of water. On very rare occasions, there have been cases where a small slug of water was trapped in piping during startup without anyone knowing it. This slug of water was isolated from the hot oil by closed valves, so its temperature was too low to cause boiling. Unfortunately, in the event of a process operator opening those valves (for whatever reason), the trapped slug of water immediately contacts 450 deg F hot oil, causing an extremely rapid evaporation rate, and resulting in an explosion. And yes, this has indeed happened. So the answer is: if the water has access to enough heat, at a high enough temperature, it will indeed instantly vaporize.
Best Answer
At high enough pressure you can keep water as a liquid above 100°C. With even more pressure you can even keep ice above 100°C. Similarly you can boil water at room temperature with a low pressure.
The phase diagram of water shows what state it is in at any given temperature and pressure.
edit: To answer Phil's question. The very steep vertical line between the blue(solid) and liquid(green) at 0°C shows why pressure isn't the reason ice skates have a film of water to slide on. You would need to increase the pressure to a few kbar to get liquid at even slightly below freezing. See how far vertically you would need to go to hit the green area starting to the left of the vertical line at 0°C
It is actually the friction between the blade and the ice that creates heat which melts the surface.