Should the moon vanish there would be less of a gravitational force being applied to the Earth as well as its inhabitants. As for the tides, they would still exist (due to the sun's gravity) however to a much less of a degree.
It should be possible to "blow up" a planet with our current technology. The more difficult question is whether or not we have the necessary supply of that tech to blow a planet up. I personally would not like to experiment this on any neighboring planets though.
Is it possible to speed up or slow down the speed of a planets rotation by attaching jet engines (or some other device) along the equator of the planet?