1. There really isn't a separate thing called heat energy. Given enough time, energy has a way of distributing around in a random-looking way among small-scale modes. These include the motions of atoms and molecules. When energy that's already gone into small modes drifts from one region to another, we say that heat is flowing. So the real question is not why heat gives motions of little things (true by definition) but why energy systematically flows from large-scale modes to small-scale ones. That follows from the second law of thermodynamics (entropy maximization), the tendency of nature to wander through all possible allowed quantum states. We don't yet know ultimately why the second law is true.
2. Thermal motions include all the parts. In a solid there are thermal sound waves running around, shaking the nuclei.
3. Your "if" clause isn't true, so I guess this question is moot.
4. The molecules in a liquid are held together by forces, most typically the vanderWaals forces. That means it takes some energy for one to pull away and enter the gas phase. Of course they do occasionally get that energy, so liquids do evaporate. The hotter they are, the more likely they are to have enough energy, so the faster they evaporate. If the molecules can escape into a huge volume, the liquid will entirely evaporate. If they're somewhat more confined, the rate of escape will be balanced by the rate of return, and the material will be part liquid and part gas.
How do you calculate how much ends up in the liquid and how much in the gas? The ultimate principle is again the second law of thermodynamics. The molecules and their energy distribute in a way to maximize the net entropy.
(published on 06/04/12)