we've had a death star project for quite some time now, it's been hushed up though, and it was designed to take care of any meteors or asteroids but I suppose it could be used for other purposes, probably won't be the last time they use something else not for what it was intended for, this is the USA after all.
1. how are you going to heat an asteroid to liquid in the middle of space? it's damn cold out there... 2. even if you could, inserting a frozen comet into a liquid asteroid would create an explosion of epic proportions, destroying your death star and anything in proximity. 3. mirrors don't grow on trees, man. and coordinating all those things? clusterfuck... 4. what? 5. ...
Space being cold is not an issue. Space, being a vacuum, is a great insulator; that is, when you heat an object, it accumulates heat very fast because the heat can only escape as radiation and not by convection or conduction.
1) constant heat builds over time. Focusing the sun can make it x times hotter. 2) the rate if expansion can be controlled by the size of the ice. 3) no, the mirrors will have to be built... Job security. (improving the economy) 4) you will need to control the laser, the solar laser. A focusing crystal is the ideal tool to use for this. Also you will need internal compartments for this laser. (read the live free or die series of books by John ringo. It's his idea)