Why does an increase in pressure raise melting point? I've heard that if you increase pressure in a system that has ice in it, then its melting point will increase. However, I would've thought that it would decrease the melting point as the ice would heat up more easily due to the air molecules colliding more frequently with the surface of the ice, so the kinetic energy is transferred to heat  ? 
Also, is the boiling point of water also increased?
 A: 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. 
A: Higher temperature will make the molecules jitter more and they tend to separate, melt at sufficiently high temperature. Increasing the pressure will counteract that separation, hence the temperature at which the material melts should increase as the pressure goes up.
A: For solids which expand on melting (eg., paraffin wax, silver, gold, copper), increase in pressure increases the melting point i.e., pressure applied is directly proportional to melting point as increase inpressure opposes expansion.
