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.
A thermometer works by having some property that depends on the thermometer's temperature in a well-defined way. You put the thermometer in contact with the system you actually care about and wait until the thermometer is in, or near, thermal equilibrium with your test system. Then the thermometer's properties measure its temperature.
But: the energy required to change the temperature of the thermometer must come from (or flow into) the system that you're measuring. So the thermometer doesn't quite report the temperature of your original system --- it reports a sort of weighted average of the starting temperature of the system and the starting temperature of the thermometer, where the weighting is the heat capacity. (Note the difference between specific heat capacity, a property of a material, and total heat capacity, which is proportional to mass.)
So: imagine that you've invented a new type of thermometric material, which for some technical reason has to begin your measurement really hot, red-hot, before it comes into equilibrium. You want to use this device to measure the temperature of a big pot of water that's sitting on your stovetop. Which would give you a better estimate of the water's original temperature: if your thermometer is a little grain-of-rice bit of material? or if your thermometer occupies as much space in the pot as the water you're trying to measure? Relate these to heat capacities and you'll have your answer.
Best Answer
Heat is not a property of a system. Heat is a process function. Temperature is a property of a system because is a state function. For instance, the state of a simple gas is given by temperature, pressure, and composition $(T,p,N)$.
Temperature is defined as the inverse ratio of variation of entropy $S$ to changes in internal energy $U$ $$T \equiv \left( \frac{\partial S}{\partial U} \right)^{-1}$$ This is the thermodynamic concept of temperature, which is more general than the kinetic concept that you are considering. Regarding your question, part of the energy given as heat is used to break the bonds and when are broken if you continue supplying energy this will increase the kinetic energy of the molecules.
Moreover, kinetic temperature is not the average of the kinetic energies of all the molecules of the object. This average of kinetic energies is the average kinetic energy. The kinetic temperature is defined as $2/3$ the average internal energy per number density.
At the other hand, heat $Q$ is defined for a given process as the internal energy interchanged which is neither work nor due to flow of matter $$Q \equiv \Delta U - W - U_{matter}$$ Notice that internal energy is a state function and $\Delta U$ denotes the difference between the initial and final energies, but heat is not a state function and this is why we write $Q$ instead of an incorrect $\Delta Q$.
The concept of process function is most easily understood with the example of a lake. A lake has some amount of water, and this can change by evaporation and raining. You can count the amount of water added to the lake by some raining process, but the lake itself does not have any amount of "raining" or evaporation" only some amount of water. Similarly a thermodynamic system has internal energy but has not heat or work.