What happens if that laser misses its target? Does it hit the closest building in its LOS?
It might. depends on alignment. it would be rare for it to lign up on a building as well as the target. this is a boost phase weapon at least as it is presented. however there is no reason it could not fire at a weapon in all stages of flight. technically. also it relies on adaptive optics in order to propagate through the shimmery atmosphere. So it's beam is focused a certain way in order to reach the target and still be able to damage it. it is likely that the beam would therefore attenuateto some degrtee between the miss and an accidental hit; probably quite a lot. the adaptive technique would cause the beam to decohere and antennuate at other ranges and atmospheric conditions.
Correct, and besides the beam director would shut off the laser in case of a miss.