The Moon is kinda far from the Earth. I don't know, man. Also why do all of the moons from other planets have names when Earth just has "The Moon"?

all has to do with the velocity imparted to lunar dust/particles by the blast - do they escape Moon gravity and end up on a trajectory to enter earth's gravity?

Not in any significant way unless it was large enough to obliterate the moon entirely then it migh be big enough to have a noticeable effect.

They should nuke Mars's ice caps

No.

Maybe

''the long-term effects of a nuclear weapon detonating in space, that radioactive material falling into the atmosphere isn’t going anywhere. In fact, it dissipates and spreads around the entire planet. Those extra molecules imparted by that nuclear blast can now be accurately measured as a means of dating things, such as trees, wine, and animals. While that may be one useful thing about nuclear testing in space, that certainly doesn’t make the whole process any more sensible''.

https://www.scienceabc.com/eyeopeners/happen-nuke-exploded-space.html

This Is The Story Of How America Once Thought About Nuking The Moon.

https://www.forbes.com/sites/robinandrews/2018/08/06/this-is-the-story-of-how-america-once-thought-of-nuking-the-moon/

Yes.