Interesting kid Mary Anne. Be sure to thank him for the questions for me. Kind of fun and challenging so here’s how I look at it.

If one isn't on Earth anymore, if one is in fact on the Moon or Mars, would one still unearth things? Would one dig in the earth, or would it be called the moon? Or would we just call it "dirt"?

I would say when digging you wouldn’t unearth dirt on the moon or mars. One would probably refer to excavating the soil or dirt. Now, if you brought some Earth soil to Mars then I would call it earth.

If earthworms were brought to Mars to use in the greenhouses, would the second generation be marsworms?

I would say they are earthworms on Mars, after all they came from earth and as long as you don’t breed them with a worm from another planet they would remain earthworms.
I think it would be sad to rename a breed of dog or any animal because it was born on Mars.

Good luck!