According to Newton's Third Law of Motion, the Earth moves every time we take a step. Not so you'd notice, of course.
As for the battleship question, it was photos like that one that prompted people to make that mistake in the first place. Sometimes so much water is pushed away by the blast that it looks like the ship is moving and leaving a wake.
That link is from Tony Digiulian's NavWeaps site, and it goes to the old Warships 1 site by Tony D and Guy Derdall. I used to post there a lot. Look at some of the other Technical links; they also debunk the '45-knot nuclear carrier' myth.
Oh, never mind, I'll get it for you:
http://www.navweaps.com/index_tech/tech-028.htm