Wouldn't the force behind him jumping off it, not to mention the giant hole now inside it, force the moon at least slightly out of orbit.. causing some destruction by all the water moving and what not?
scientifically, no. As the moon has far less of a gravitational field to it, Saitama wouldn't have to do much in order to go back to earth. and if it did push the moon back, it would only be at most 50-100 m which is really not that much distance
Fair enough, I guess. I do recall the manga did have him question, mentally, how much gravity difference there was. Don't think they did that in the anime, but he probably calculated it to have the least impact or something by instinct.
But they missed their chance of having a Sailor Moon-like character scolding him for breaking the moon as a joke.
No, solid heavenly bodies are more like a giant dirtball held together by gravity than a solid piece of rock, hence the moon would split into fragments from impact before it got significantly out of orbit.
Hence the bad strategy of nuking giant asteroids.