The pressure in three different directions is indeed independent for materials that are composed of cubes or other fixed shapes. But these materials are called solids, not liquids.
By definition, a liquid is a material without any regular crystallic or otherwise periodic structure. A liquid is composed of randomly arranged molecules that are as close to each other as the repulsive forces allow – the latter property distinguishes liquids from gases. Liquids and gases are two subgroups of a larger group called fluids.
When a milliliter of liquid is at rest, it means that in this milliliter of material, the molecules are randomly ordered and randomly moving so that their center-of-mass remains at rest. But when it's so, the only "force-related" quantity by which one milliliter of this liquid differs from another is the density or any function of it, such as the pressure. Because there's only one density, there's only one pressure.
Because liquids are not composed of fixed cubes but of chaotic molecules. a new molecule added to a volume of liquid with excessive pressure may escape in any direction. Whatever direction it chooses, the density (in molecules per unit volume) will be reduced to the appropriate value so the pressure may drop, too.
The independence of the fluids' pressure on the direction is known as Pascal's law and you may read an independent explanation at Wikipedia:
http://en.wikipedia.org/wiki/Pascal%27s_law
All first order phase transitions have a change of volume. With different pressures you need to consider the sign of the work $P\Delta V$ that needs to occur during the phase change. If $\Delta V$ is positive, the phase change will occur at a higher temperature for higher pressure. If negative, the phase change will occur at a lower temperature.
(Note that how the temperature is changed, or how fast, has nothing whatsoever to do with thermodynamics - that is a kinetic issue and does not impact the relative free energies of the various phases.)
Now, for boiling water, the molar volume of steam is larger (by a lot) than the molar volume of water at the boiling point. Increasing the pressure results in higher boiling points. This is the basis of pressure cookers, superheat steam engines, etc. On the other hand, ice has a lower molar volume than water (it floats), so increasing pressure leads to a freezing point decrease.
Best Answer
This is because of the Clausius-Clapeyron equation $$\frac{d \log T}{d \log P} = \frac{P \Delta V}{L}$$ where $T$ is the temperature of the phase transition, $\Delta V$ is the change in volume, and $L$ is the latent heat. The water/gas transition has an enormous $\Delta V$ because gas is much less dense than water, so $dT/dP$ is large. The water/ice transition has a $\Delta V$ about $10^{-3}$ as big, so $dT/dP$ is small.
Intuitively, there is some 'cost' $L$ to be paid doing the phase transition, and usually most of it is paid by thermal energy. But if the volume changes during the transition, the $P \Delta V$ work can also help, lowering the necessary temperature. So it makes sense that $dT/dP$ depends on the ratio of these two contributions.