Is liquid water in equilibrium with water vapor at room temperature?
It depends on the circumstances, but mostly not

Under most normal atmospheric circumstances water in an open vessel is not in equilibrium with water vapour in the atmosphere. We know this because water usually evaporates, albeit slowly. 

One way to measure whether water is in equilibrium is to measure the humidity which is a measure of the maximum concentration of water vapour under the prevailing temperature and pressure. Humidity is mostly less than 100% which means that the air could have more water vapour in it before the vapour spontaneously condenses. This isn't always true, which is why we get fog and why some countries are unbearable to live in.

At normal temperatures the equilibrium between water and water vapour is established slowly. Any vessel containing water that is open to the atmosphere will try to establish that equilibrium, but that won't be complete until the air is 100% saturated with vapour (humidity=100%). And you'd have to saturate the entire atmosphere in the room or, if the room is open, the entire planet's atmosphere to achieve equilibrium.

That this doesn't happen quickly is obvious. But it can happen locally (we call that weather). It it obviously not true across the whole planet at the same time or the planet would be called "sauna" and not earth.

The point is that the atmosphere is rarely 100% saturated with water vapour, so nothing open to the atmosphere will normally be in equilibrium with liquid water. A close bottle left long enough might establish equilibrium (with 100% humidity) inside the closed bottle, but any exchange with the surrounding air will lower the humidity and the equilibrium might take some time to re-establish.

The key to understanding this is that the kinetics of establishing the equilibrium are slow and the normal atmosphere is not saturated with water vapour. This means that the air above a glass containing open water will have a slightly higher concentration of vapour than the air surrounding the glass but with not be fully in equilibrium because the room surrounding the glass is big. And the atmosphere around the room is even bigger and clearly isn't in equilibrium (we have weather and wind to demonstrate that). The only systems where equilibrium will be established are closed ones which have been left a reasonably long time. atmospheric exchange causes the first; slow kinetics the second of these constraints.