I don't know the answer to your second question, but for the first question I would say that the laser pulses are short due to the high power (there is no laser system in the world that would manage to output TW or even GW laser beams continuously). But theoretically, I see no problem using a continuous wave laser to ionize a solid target as long as you reach the necessary field strengths to ionize via tunnel ionization to get the plasma started.
Since you mentioned that the targets Coulomb explode, I'll just touch lightly upon that also. The reason for this explosion is that the electrons in the plasma are much more mobile and can be accelerated to escape velocity from the positive ions by the laser. Then once the electrons have escaped the ions' mutual repulsion cause the Coulomb explosion you mentioned. There are several acceleration processes, but usually the rapid rise in intensity of a short pulse is involved in accelerating the electrons. Therefore a continuous intensity laser might not be able to make the target Coulomb explode, even though it might be powerful enough to ionize the plasma. (Once you have created a solid density plasma though, it will explode through simple hydrodynamic expansion into the surrounding vacuum.)