When is it correct to capitalise 'earth'?

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.

1. A handful of earth.
2. The earth under this house.
3. 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.

According to the Wikipedia Manual of Style, it appears your examples should be as follows:

1. A handful of earth.
2. The earth under this house.
3. The earth beneath my feet.
4. What on Earth?
5. The heavens above and the Earth below.
6. The Earth moved.
7. We returned quickly to Earth.

Also, I believe 5 and 6 could have either capitalization, depending whether you mean the planet Earth, or just the dirt below your feet.

1. A handful of earth.
2. The earth under this house.
3. The earth beneath my feet.
4. …
5. The heavens above and the earth below.
6. The earth moved.
7. …

In the examples above, earth should be lowercased. In all the examples cited, earth is synonymous with the following common nouns: soil, land, dirt, terrain, ground, and world. In addition, the expression in number 6. is idiomatic but it doesn't normally refer to earthquakes, it means that something, often a sexual experience, was highly pleasurable.

For example,

1. A handful of soil/dirt
2. The ground under this house
3. The land beneath my feet.
4. …
5. The heavens above and the world (or ground) below.
6. The terrain moved OR The world moved
7. …

I feel example 4. could go either way, Googling the phrase does not really help to clarify.

4. What on earth?
   What on Earth?

On the first page of Google, earth is capitalised 14 times out of 20 but many of the instances cite the name of an American TV series, the titles of podcasts; events and shows; science exhibitions; they also form the name of several websites, e.g. What on Earth is Happening?; a BBC science page, and titles of books.

Cambridge Dictionary says of the phrase

what on earth

used for showing surprise:

What on earth is going on in there?

Lexico, formerly Oxford Dictionaries, has an entry for on earth, lowercased

on earth

Used for emphasis, especially in questions and negative statements.

‘who on earth would venture out in weather like this?’
‘So my question is, how on earth do you keep tabs on all of the available programming?’
‘So why on earth does he want to make yet another death-defying walk in the Grand Canyon?’

Dictionary.com uses lowercase for the following idioms

• move heaven and earth.
  on earth, in the world:
  Where on earth have you been?

Merriam-Webster includes another idiom, earth is still spelled with a lowercase "e"

(a) heaven on earth

and when earth used to be flat,

from the four corners of the earth
People came from the four corners of the earth to see the sight

The following definition by Macmillan Dictionary

on earth

1. used for emphasizing that someone or something is the best, worst, biggest etc in the world
   The Great Wall is the largest man-made structure on earth.

2. nothing/nowhere etc on earth used for adding emphasis to negative statements
   Nothing on earth could get me to speak to her.

But in one dictionary, TFD, the word earth is capitalised

go to the ends of the Earth

To do all that one can in an attempt to achieve something.
I would go to the ends of the Earth to help my children.

The one sentence where I would capitalise earth is the last one

7. We returned quickly to Earth.

The MLA Style Center says

We usually lowercase sun, moon, and earth, but, following The Chicago Manual of Style, when the does not precede the name of the planet, when earth is not part of an idiomatic expression, or when other planets are mentioned, we capitalize earth:

 The earth revolves around the sun.
 The astronauts landed on the moon.
 The space shuttle will return to Earth next year.
 The four planets closest to the sun—Mercury, Venus, Earth, and Mars—compose the inner solar system.

I'm surprised by their first example, it's clear that the term earth refers to the planet on which we live, and it cannot be replaced by soil, land, ground, dirt etc. Even though it is preceded by the definite article the, it does not dissuade me from wanting to capitalise that all-important "e"

Lexico provides the following example sentences

• ‘The discovery suggests that life could exist on planets very different from Earth.’
• ‘The course of life on planet Earth might even turn out to be described by such a picture.’
• ‘Mercury is also the only planet other than Earth that has a global magnetic field.’
• ‘Imagine a perturbation of the Earth's orbit big enough to change the size of the sun in the sky.’

In none of the above examples could the word earth be substituted with any of counterparts without affecting or changing its meaning substantially.

Interestingly, in the article Scott Kelly’s Year in Space, published in February, 2016. The New York Times journalist cleverly avoids using the article with Earth.

• The International Space Station zips around Earth at more than 17,000 miles per hour, or once every 90 minutes.

• Of course, on the space station, Mr. Kelly was never more than about 250 miles from Earth.

• Water is heavy and expensive to transport from Earth, so for efficiency, water is continually recycled.

• Whatever Earthly things Mr. Kelly may have been missing during the mission, the Internet was not one of them.

I would write Earth when referring to the planet where we live because it is a proper noun (like Mars, and Venus, for example).

Looking at the New Oxford American Dictionary, it's reported that

The earth is the third planet from the sun in the solar system, orbiting between Venus and Mars at an average distance of 90 million miles (149.6 million km) from the sun, and has one natural satellite, the moon.

The definition for earth as a noun starts with

1. (also Earth) the planet on which we live; the world: the diversity of life on earth.

As for the example phrases you wrote, I would write them as

A handful of earth.
The earth under this house.
The earth beneath my feet.
What on earth?
The Heavens above and the Earth below.
The earth moved.
We returned quickly to Earth.

I wrote earth on the fourth phrase because it's a way of saying, which has a different meaning from What is happening on Earth? I am actually not 100% sure on how I would wrote earth in that case.
In the earth moved I wrote earth because the word is not referring to the planet (I suppose it is the sentence you say when there have been a quake).