Would water flow out of this container? Consider a container that is partially submerged in water as shown in illustration at the bottom of the question. A, B, C and D are valves, that are currently closed. The container is full of water, because before the experiment we pumped air out of it through valve D (let's pretend that the air pressure outside the container can push the water up in the container above the valve A under these circumstances - also the container is secured in its position as pictured). 
The experiment proceeds as follows: we open all four valves at the same time and observe what happens.
The question is - will the water during experiment flow through valves A B or C out of the container, or will the air rush into the container through these valves?

In my opinion the latter is correct - the air will rush in through all valves - I imagine the situation is the same as having low pressure zone below the container. However, the water surrounding the container makes me a bit unsure - does it affect the results of the experiment?
 A: Because the water in the system is assumed to be static and connected, we know the pressure at any particular level is the same.  So the pressure near the bottom of the vessel (below valve D) is equal to that at the surface outside the vessel, which is atmospheric.
As you go up the vessel, the pressure inside decreases with the pressure lapse rate of water, $\rho g h$ or $9807 \text{Pa/m}$, while the pressure outside is nearly constant.
So the internal pressure at any point above the surface is less than atmospheric.  The action of opening any valve will allow air to rush in, equalizing the pressure.
After opening a valve, the water will start moving and you get a dynamic system with changing pressures inside the vessel.  Analyzing the motion of the water at that point becomes more complex.
An alternate way to visualize this is to imagine how the situation was created.  The pressure is initially atmospheric, but lowered as air is pumped out.  The pressure at any level inside reduces until water reaches it, and is constant from then onward.  So every point inside will be below atmospheric.
A: My initial answer, as pointed out by @SV, was incorrect. My apologies! I wrote:

"You are probably imagining that there is a "suction" operating due to the weight of the water, but in fact there is only positive pressure both inside and outside the container, and water flow is simply due to the pressure difference.
Consider the pressure difference between the inlet and outlet of, e.g., valve A.  At the moment when all valves are opened, there is a higher pressure at the inlet (the container side) of valve A than at the outlet (the air side).  So, water will flow out of valve A until the pressure is the same at both its inlet and outlet.  The same is true of valves B and C."

I tried the experiment, and indeed water did not flow out of A, B, and C.  Here is my explanation:
Water pressure inside the container decreases linearly from ambient air pressure at the external surface of the water (0 feet), up to zero at a height of about 33 feet.  Air pressure, on the other hand, decreases very slowly with height above the water surface.  So, the pressure inside the container is less than the air pressure at every height.  Since the flow direction is from high pressure to low pressure, when the valves are opened air will flow into the container through all valves D, A, B, and C.
