If you compress an ideal gas adiabatically, the average kinetic energy of the particles will increase because collisions with the moving piston will increase the kinetic energy of the colliding particle. For an ideal gas, the intermolecular distances have no bearing on the internal energy.
More specifically, suppose the piston is moving inwards at velocity $u$ and a gas particle hits the piston wall at velocity $v$. Because the piston is much more massive than the particle, you can assume that it gets completely reflected, so the particle will come away with velocity $v+2u$. In practice, $v$ is much bigger than $u$, so this amounts to only a small increase in $K_E$ per collision, but the piston is moving slowly and there will be many such collisions over the course of the compression.
If the compression is being done isothermically, then you are assuming that any excess kinetic energy is being dumped into the heat reservoir. The flow of energy is then from the moving piston to the kinetic energy of the gas particles through collisions with the piston, and from there to the reservoir, and this is assumed to be at such small increments that the mean kinetic energy does not change. Again, for an ideal gas, the intermolecular distances have no bearing on the internal energy.
Regarding water ice, the process of melting is obviously complicated and you cannot simply wrap it up into a simple function of the mean interparticle distance. There are complex interactions between the molecules and these depend quite sensitively on their relative orientation, with the end result being the crystal structure of ice. Your question is, basically, "what is it about the molecular interactions in water that makes the lowest-energy crystal form have the structure it does?", which is obviously a much more complicated question in and of itself (though, of course, it's very well studied).
You misunderstood le chatlier principal.
By saying that" its effect to reduce", it meant causal effect is to be reduce. So, here he meant on increasing pressure since volume decrease , here he meant to move from more volume towards that direction which has less volume, where increased pressure has less effect.
He hasn't decreased volume , decrease in volume is result. He increased pressure, so, he wants to decrease pressure. when he apply pressure on ice it is converted into water and pressure is decreased. It is like an inflated ballon. If balloon has air filled due to more air, baloon apply more force on air. Where as if we have something like nail in it, negligible pressure is applied on it.
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Basically a "molecule" of water cannot heat up ice.
I think what you are trying to say is, how does heat transfer take place on a molecular level? If that's the case, then its something like this. In the interface between water and ice, water molecules are moving, while ice molecules are static. on contact, some molecules of ice acquire velocity (due to no binding forces in one direction, and cohesive forces towards liquid water molecules). as a result, surface molecules of ice start acquiring velocity, hence changes state from solid to liquid (simply, ice melts). and due to conservation of energy, an equivalent amount of kinetic energy (macroscopically, heat) is lost by water. hence water cools a bit.
Of course there may be better ways to explain this, but i tried my best. hope you understand :)