Why does fluid compression cool it up, unlike solids? Why is it that solids on compression [As in striking a hammer etc.] heat up,
but liquids and gases on compression [Pressurizing liquids causes them to freeze or gases to liquify] cool down? 
 A: Your basic premise is wrong.  Compressing things makes them heat up.  With gasses this is even nicely predictable over a wide range.  Look up something called the ideal gas law.
If you compress a gas enough then it will undergo a phase change and turn to liquid.  This will release even more heat than just compressing the gas because the gaseous state of a substance has more energy than the liquid state.  This is what the heat of vaporization is about, and why you have to heat a liquid to make it boil (change phase to its gaseous state).
Steam heat systems work on the reverse principle.  The steam condenses on the inside of the pipe, thereby releasing a lot of heat, which heats the pipe, which heats the room.  The condensed water returning to the boiler contains less energy than when it left the boiler as steam.
A: Have a look at the phase diagram of water as an example. It shows the state of matter that water is in at different temperatures (horizontal) and pressures (vertical).
Compressing (that is increasing the pressure) in the gaseous phase will increase the boiling temperature. So if you pressure water vapour, but keep the temperature constant, for example by allowing heat transfer to the outside, then at some pressure the gas will liquefy. It is a misconception that the boiling point is always at a fixed temperature.
For liquid water the freezing point around the normal atmosphere pressure (horizontal red line in the diagram) is nearly constant. So solidifying by compression will either require a reduction in temperature or a very high pressure. In fact increasing the pressure to around 1kbar will reduce the melting point and only over 6kbar the melting point becomes higher than usual.
Compressing a solid might also change the phase, but it will stay a solid phase.
Now to your proposed experiment, striking a solid with a hammer. The kinetic energy of the hammer has to transfer to the solid as work or heat. Heat will result in increased temperature, while work is associated with change in volume. Since solids are generally hard to compress, the change in volume and thus the work fraction will be small. Most of the energy will result in heat. Since the solid as a whole is not confined to some space and the pressure is released again very fast, the solid will relax again and release also the compression work as heat. After some time, all the heat will have transfered to the environment and the initial temperature will be reached again.
Now compressing water vapour. Here it is not clear whether you are talking about compressing the vapour fast and/or thermally isolated (adiabatic) or slow, such that heat can transfer to the environment and temperature effectively stays constant (isotherm). For the isotherm case I already gave you the result above: The boiling point depends on temperature, but also on pressure. Compressing will at some point lead to liquefaction at without change in temperature. In the adiabatic case pressure as well as temperature will rise. It is not clear from the phase diagram whether this will result in liquefaction or not. However liquefaction releases heat, which would further heat the substance, which is not preferable. So there won't be liquefaction.
For isotherm and adiabatic compression of the liquid I already showed you that for typical everyday pressures no solidification will occur, since the melting point is initially decreasing with pressure.
In summary your conception that a change in phase requires a change in temperature is wrong, because a change in pressure may be enough already.
