How does water pressure manifest itself on the molecular level? tl;dr: I kept hearing people say "water is incompressible... there's just as much water mass in a cubic meter at the bottom of the ocean as at the top". Is that true?
Answer: this is false. Water is totally compressible. There is, in fact, more water mass in a cubic meter at the bottom of the ocean as at the top. It's just that huge changes in water pressure can come from tiny changes in water density.
On the molecular level, under higher pressure, water molecules get closer together, and their atoms' electron clouds repel each other, which is basically how solids and gasses also compress.

How does water pressure manifest itself on the molecular level?
Second, related question: does water density relate to, or cause, water pressure? Everyone I talk to dismisses the idea that water density substantially affects pressure, and emphasizes that water is nearly completely incompressible.
Bit I wonder if that "nearly" is a small percentage, but a big deal.
Similar question:
Molecular explanation of pressure in water at depth
Here's my scenario: we have a hundred-gallon cylindrical water tank, and we poke three holes into it at different heights, stick tubes in the holes, and watch as the water comes pouring out of each tube, with the highest tube producing barely a trickle, and the lowest providing a hard spray. So far, so good, this is a standard grade school experiment.
But lets say we cap each tube from the outside. Lets assume, further, that the tubes are made of incredibly strong and incompressible material, and very narrow, so we can consider only the horizontal pressures and ignore vertical pressures inside of the tube. Now the water pressure will build up inside each tube, ready to spray when we remove the cap. 
I'm wondering about the physical state of the molecules closest to the caps. When you remove the caps, those molecules that are now exposed to air are identical within each tube, in terms of velocity and momentum. And they are nearly identical, though not absolutely, in terms of density, which means the next layer of water molecules next to them are essentially the same distance away in each of the tubes.
And yet, once the caps come off, these outermost water molecules will begin to accelerate at very different speeds. The reason, as I understand it, is that there is greater force being applied to all of the nearby water molecules because of the distributed force of gravity on the water higher up in the tank.
So my questions are:
a) if you capped the tubes on the inside before uncapping than on the outside, would the water still spray at different velocities?
b) does the greater density under higher pressure have any effect on the pressure force that nearby molecules feel?
c) how does force transfer from one molecule to an adjacent one? Through nuclear proximity, which engages the weak nuclear forces? Or electromagnetic forces? Or some other force that doesn't want atoms to get too close to each other?
d) if b) is false but c) is true because if proximity, how can greater density NOT affect pressure? 
e) Isn't the whole reason that water is difficult to compress that when you bring water molecules into such close proximity, your kinetic energy is converted into potential energy, which is stored in the greater resistance on the atomic level? 
f) is water pressure a local phenomenon, like the way gases exhibit pressure by colliding more often, or is it a macro phenomenon, exhibited in the readiness of so many water molecules to for into a space?
g) how can a massive collection of molecules, like in a thin, tall cylinder of water, all be subject to a force applied from above, if not through some observable physical transition like an increase in density? Is there a time consuming process, like a shock wave, by which a newly applied force is effectively communicated to all of the matter which is subject to it?
h) a way of restating g) is to assume that at the same moment that we uncap the tubes in the original example, we add another hundred gallons of water to the tank, by removing a separator at the top of the original tank. How, and when, will the acceleration of the water in the tubes reflect this new gravitational pressure? Surely not immediately, or we've just discovered faster-than-light communication. So then presumably, some information or physical change must spread through the water which triggers a shift in behavior to reflect the greater pressure. Won't this correlate neatly with an increase in density, even if that is inconsequential? What form does this meaningful transition take, if not an increase in density?
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Edit: so I've been reading responses to this and other questions. Here's what I'm coming to understand, I * think *. (But note that at least one incredibly well educated scientist I know disagrees with me.)
a) yes, basically. but only slightly different velocities, and all lower than before; you would need the pressure from the rest of the water, and the arrival of that water, to make the flow harder and more continuous. But the density before the capping would be preserved, and that contains significant potential energy.
b) yes -- density is a necessary component of water pressure, and it is essentially through increasingly difficult increases in density that water obtains more and more pressure.
c) force transfers via periodic, somewhat random encroachments of one atom's space onto another's, and one molecule's onto another's. This forces electrons to get closer and thus to engage electromagnetic repulsive forces, and within molecules, resistance of some component atoms to other molecules' atoms makes the molecule deform shape and resist this deformation electromagnetically.
d) n/a
e) yes
f) local. that "readiness" to flow is made manifest in the form of the increased frequency and persistence of atomic encroachments on the space of other atoms. If you could magically make a thin wall of water molecules encroach on their neighbors aggressively, the molecules nearby would act precisely as though there were tons of water above that wall experiencing downward gravitational force.
g) they can't, they need to be made more dense and deformed, a pattern which travels like a wave through the water.
h) it will correlate precisely with the increase in density because it is density, and the continued agitation of molecules in a dense arrangement, that actually expresses, and stores, the changed pressure.
 A: I will focus on just a little bit of one of your questions - the relationship between compressibility, density and pressure - and per my comment, recommend that you narrow down the scope of your question.
As you know, in a gas we experience "pressure" because molecules hit the walls of the containing vessel. When I double the number of molecules in the same volume at the same temperature, I double the number of collisions (each imparting on average the same momentum) and thus double the pressure - this is the familiar ideal gas law.
Now when the size of the molecules becomes a sizable fraction of the volume, the rate of collisions goes up. Imagine a pingpong ball between two walls. If the distance between the walls is large compared to the size of the ball, the time for a round trip is inversely proportional to the size of the ball; but as the distance approaches the size of the ball, the rate of collisions goes up rapidly.
I think a similar thing happens with "nearly incompressible" liquids: there is a small amount of space between the molecules, but they are permanently bumping into each other and into the walls of the vessel. As you increase the pressure, they bounce more frequently as they have less far to travel before they collide with another molecule (or the wall).
All this is still treating the liquid like a non-ideal gas. In reality, not only do you have the finite size of the molecules, but also attractive forces between them. Both these things make the picture a bit more complex than I sketched. But I would say that the above reasoning nonetheless applies (with caveats).
As for the experiment you described with stoppers on the inside or outside - there are other things going on there as you go from the static (no flow) to the dynamic (flow) situation - the water needs to accelerate before it will flow out at a certain velocity. But I think all that should be the subject of another question.
A: Your intuition about pressure is coming from PV=nRT, the ideal gas law.  In that, more molecules (n) or smaller volume (V) means a proportional higher pressure because there are more molecules bouncing per second off a given bit of wall:  More collisions per second per cm^2.
But that's the ideal gas law. Liquid water is far from ideal, and it's certainly not a gas.
With only unusual exceptions, liquids and solids have their molecules continually in contact.  For small compressions, not enough to change the phase, the compression is compressing the molecules themselves. When you compress liquid water, you're trying to compress the electron "clouds" that form the orbitals and bonds.  Electrons don't like being forced closer together...
