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.
Your suspicion is correct, the Bonnor-Ebert mass is indeed describing the same instability condition as the Jeans instability. This is the gravitational instability, which is tied to the competition between self-gravity and internal pressure. The only reason they are given different names is that they represent two different ways to arrive at this instability condition, which end up giving compatible results.
To derive the Bonnor-Ebert mass, one considers equilibrium solutions to a spherically symmetric configuration of self-gravitating gas in hydrostatic equilibrium. These are solutions to the Lane-Emden equation. One can then consider the stability of normal modes in these solutions to perturbations. For an isothermal equation of state, the fundamental ("breathing") mode is unstable whenever the mass of the sphere exceeds
$$ M_{BE} = 1.18 \frac {c_s^3}{\rho_0^{1/2} G^{3/2}}$$
where $c_s$ is the isothermal sound speed and $\rho_0$ is the central density of the sphere. In other words, spherical configurations of gas with isothermal equations of state that are initially in equilibrium are unstable to collapse when perturbed if they are more massive than $M_{BE}$. This analysis can be extended to gas with non-isothermal equations of state.
To derive the instability from the Jeans point of view, one can instead consider traveling-wave perturbations (sound waves) traveling through a self-gravitating, homogeneous medium of density $\rho_0$ and isothermal sound speed $c_s$. Through linear analysis of small amplitude waves one can derive the dispersion relation for these waves and conclude that when the wavelength exceeds a critical length,
$$\lambda_J = \sqrt{\frac{\pi c_s^2}{G \rho_0}}\, ,$$
the amplitude grows exponentially in time. So structures that have lengths larger than this are susceptible to collapse when subjected to perturbations. Again, this was all done in the isothermal case for simplicity, but can be generalized to other equations of state.
To see how these two analyses relate to each other, consider the mass enclosed within a sphere of diameter $\lambda_J$ and uniform density $\rho_0$. You'll see that it is the same as $M_{BE}$ to within a factor of 2 or so, which shouldn't be too worrisome given the differences in the initial configurations (an equilibrium sphere for $M_{BE}$, and a uniform medium for the Jeans analysis).
For a detailed discussion, see e.g. chapter 9 of this textbook. The primary application there is star formation, but the physics is general and may be applied to other situations.
Best Answer
The answer lies in something called the virial theorem and the fact that the contraction of a gas cloud/protostar is not adiabatic - heat is radiated away as it shrinks.
You are correct, a cloud that is in equilibrium will have a relationship between the temperature and pressure in its interior and the gravitational "weight" pressing inwards. This relationship is encapsulated in the virial theorem, which says (ignoring complications like rotation and magnetic fields) that twice the summed kinetic energy of particles ($K$) in the gas plus the (negative) gravitational potential energy ($\Omega$) equals zero. $$ 2K + \Omega = 0$$
Now you can write down the total energy of the cloud as $$ E_{tot} = K + \Omega$$ and hence from the virial theorem that $$E_{tot} = \frac{\Omega}{2},$$ which is negative.
If we now remove energy from the system, by allowing the gas to radiate away energy, such that $\Delta E_{tot}$ is negative, then we see that $$\Delta E_{tot} = \frac{1}{2} \Delta \Omega$$
So $\Omega$ becomes more negative - which is another way of saying that the star is attaining a more collapsed configuration.
Oddly, at the same time, we can use the virial theorem to see that $$ \Delta K = -\frac{1}{2} \Delta \Omega = -\Delta E_{tot}$$ is positive. i.e. the kinetic energies of particles in the gas (and hence their temperatures) actually become hotter. In other words, the gas has a negative heat capacity. Because the temperatures and densities are becoming higher, the interior pressure increases and may be able to support a more condensed configuration. However, if the radiative losses continue, then so does the collapse.
This process is ultimately arrested in a star by the onset of nuclear fusion which supplies the energy that is lost radiatively at the surface (or through neutrinos from the interior).
So the key point is that collapse inevitably proceeds if energy escapes from the protostar. But warm gas radiates. The efficiency with which it does so varies with temperature and composition and is done predominantly in the infrared and sub-mm parts of the spectrum - through molecular vibrational and rotational transitions. The infrared luminosities of protostars suggest this collapse takes place on an initial timescale shorter than a million years and this timescale is set by how efficiently energy can be removed from the system.