Generally speaking, I would write earth when referring to soil, the ground, or land as opposed to sea. I would capitalize Earth when referring to the planet (and even more specifically, our planet; there are many earths out there). In short: if it's a proper noun, it should be capitalized.
- A handful of earth.
- The earth under this house.
- The earth beneath my feet.
I agree that earth should not be capitalized here. However, I could come up with contexts in which I would capitalize it even in these sentences, though they would be rather poetic or sci-fi (read: far-fetched).
4. What on earth?
I would capitalize Earth here. To me, this question is equivalent to "What on this planet?", "What in this world?", "What on Mother Earth?", and not to "What on soil?" or "What on land?". (If you want a comprehensive, highly scientific study, then I am happy to report that out of the first 30 Google results for "What on Earth", Earth is capitalized in 25 cases.)
5. The heavens above and the earth below.
Soil, ground, land as opposed to the sky. No need to capitalize.
Edit: as Neil Fein points out in the comments, this could refer to the planet in certain contexts, in which case I would capitalize it (cf. 1, 2, and 3).
6. The earth moved.
It depends. Did the Earth (proper noun) move around the Sun (proper noun), or did the earth (soil, ground) move because there was an earthquake?
7. We returned quickly to earth.
I agree with you that here it should be capitalized.