Heat is energy in transit. It only comes into play when one body is transferring heat to other. Feynman best explained in a video on youtube, it is just the vibrations of atoms at an atomic scale. So, if a more vibrating solid object comes in contact with solid with relatively low "jiggling", we say heat is transferred.
[Ref]: Feynman lectures on physics vol.1 ch 39,40
Pressure is the total force by a gas on walls of chamber/Area. Simple calculation gives it's physical meaning as: (number of atoms)*(average KE per molecule)/ Total inner surface area of chamber
Temperature is just (another way to write) average kinetic energy per molecule.
I don't think it's physically possible to move piston without giving energy. But suppose we have to increase pressure. How can we do it? By decreasing the surface area, or increasing number of atoms or increasing K.E.
If we somehow decrease surface area(volume of chamber, by pressing piston), very slowly so that not much force acts on particles increasing K.E, we can say Temperature didn't increase. I'd like to have comments on my statement.
3. If we increase pressure of gas-liquid equilibrium, we can easily imagine, gas will collide more and more with liquid and this tends to increase boiling point of a liquid compared to at 1 atmosphere. Similar reasoning is imaginable for your question.
4. Pressure doesn't reduce motion of particles. It is just an effect on chamber that we observe.