I'll give brief answers to your questions. If you need more detail, you should ask your questions separately.
What's the difference between heat and work at the atomic level? Isn't heat simply work between particles colliding with different momentum against each other?
Treating a substance semi-classically, one can say that at the atomic level, the atoms have a certain position and momentum. Quantum mechanically, even that's dubious because position and momentum are conjugate variables. With regard to heat and work, these don't exist at the atomic level.
Heat and work are processes, not states. Atoms don't contain heat or work. Neither do individual collections of atoms. Heat and work are measures of quantities transferred amongst objects. Objects don't contain heat or work.
Does an increase of pressure also increases the temperature of the gas?
For an ideal gas being compressed adiabatically, the answer is an emphatic yes. For anything else, the answer is sometimes yes, sometimes no. The answer depends on how much heat is being transferred into or out of the gas and on the nature of the gas. If the gas is right at the triple point (ideal gases don't have a triple point), all that compressing the gas adiabatically is going to do is cause some of the gas to turn into liquid or solid.
Excluding water and other special materials, why does a increase of pressure over a solid rises is melting point?
What your teacher told you is nonsense. Increased pressure does not decrease the molecule's motion. What increasing the pressure does do is to decrease the intermolecular distance.
The reason most substances contract when they freeze is because the bonding forces that make a substance become a crystalline solid hold the atoms/molecules closer together than the intermolecular distance at the same temperature in the liquid phase. Increasing the pressure in these substances decreases the intermolecular distance, thereby making it easier for those intermolecular bonding forces that make a substance a solid to take hold.
Water is different. It expands upon freezing. The structure of ice (ice Ih) is very open thanks to the hydrogen-hydrogen bonds in ice. Because ice expands upon freezing at normal pressures, increasing the pressure reduces the freezing point. Increase the pressure beyond about 100 atmospheres and water/ice starts behaving like most other substances. Increase the pressure beyond 3000 atmospheres and something even weirder happens. Now the freezing point drops markedly with increasing pressure. Increase the pressure beyond that and something even weirder happens: The freezing point increases again, this time very sharply increasing with rising pressure. The freezing point is over 600K at a pressure of 100,000 atmospheres.
If the pressure reduces the motion of the particles, how can the inner core have material with higher temperatures (i.e. particles with higher average kinetic energy)?
What your teacher told you was wrong.
Best Answer
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.