The moon has a mass of 7.349×10²² kg. Its mean orbital velocity is 1023m/s. So, with a perfectly-aligned impact, and perfect efficiency, to add velocity, you'd need to add energy equal to difference in kinetic energy. For 1m/s, that'd be 10²⁵J (assuming the change in mass is negligible). That's an incredible amount of energy, about 20 times greater than the Chicxulub impact (believed to have triggered the mass extinctions at the K-T boundary).
That's 1 m/s. You need far more than that. Earth escape velocity is 11200m/s. So, that would seem to need 10³⁰ J. And an impact would deliver that in well under a minute. Which would be a problem, since that's an order of magnitude greater than the gravitational binding energy of the moon.
Conclusion: impact required would fully obliterate moon.
(Moon mass and velocity, and escape velocity from http://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html. Gravitational binding energy and estimate of Chicxulub from Wolfram Alpha)
PS if moon gets a major hit, its highly unlikely earth stays 'physically' unaffected. The aftermath of the collision would leave numerous chunks of huge rocks in earth's near space and it would be almost certain that some of then then hit earth. And the effect of decent sized rock hitting the planet is better known to our long lost friends, the dinosaurs :)
But would the nights be totally dark? I've experienced a lot of nights out on a boat on a moonless night and the stars provide a (relatively) large amount of light on their own.
As for who gets hit; Earth's mass is 81 times that of the Moon so it attracts objects 81 times as strongly. Really, the Moon would only get hit if Earth already pulled an object into a collision course or near-miss and the Moon just happened to get in the way at the right time. Earth has 4x the diameter so presents 16x the cross-sectional target that the Moon does.
