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.
Glass breaks because it is brittle instead of flexible; this means that if the shape of the glass deforms enough, if a surface bends just a little, it breaks. If the shape of the bottle doesn't change, then it won't break, no matter what forces are applied to it.
In the case of a bottle that is full of water with no air, the force of the impact with the nail causes one side of the bottle to deform. But, water is incompressible, so the water stops the side of the bottle from bending more than a negligible amount in order to keep the water volume constant (this is what "incompressible" means). Now, pressure is force divided by area, so the force driving the nail is spread out over the entire interior of the bottle by the water, so no part of the glass bends enough to break. That's why the bottle would survive.
In the case where the bottle has a bubble in it, the story would be different. Air and all other gases are very compressible. So, upon impact with the nail, the side of the bottle impacting the nail would deform. The gas would compress from the transmitted water pressure, allowing the water to move out of the way of the deforming side of the glass bottle into the volume formerly occupied by the bubble. This allows the glass to bend more, resulting in it breaking.
To compare, imagine the result of the virtual experiment with the corked bottle full of water with no bubbles at 11:44. If the professor had hit the top of the cork, the bottle would have shattered. Why? In order to stop the cork from entering the bottle (to keep the volume of water from compressing), the water would have to deliver a large force to the cork to stop it. This requires a large pressure since the area of the cork face is small ($Force = Pressure \times Area$). This pressure is transmitted to all sides of the bottle, generating an enormous force since the bottle interior has a much larger area. Air pressure outside the bottle is far too weak to prevent the bottle's sides from bowing outward and breaking. This is the basis behind the hydraulic press seen from 4:37 to 7:20.
Best Answer
The effect is real.
The heat in the bowl causes the production of steam in the cavity between it and the table. Depending on the temperature, this can be a far more powerful effect than mere thermal expansion of the air. The liquid between the bowl edge and the table acts like a liquid seal for a reasonably smooth and planar table/bowl interface - liquid being held in place by surface tension.
All the above give you a "hovercraft" which has extremely low friction. Now all you need it a tiny push. This could be a small slant on the table, but the expulsion of small amounts of steam & water from underneath the bowl (in random directions) can be sufficient to push - and that will generate random motion.
I think what you observed is real - all it needs is a well fitting hot bowl and a wet table.