Name for the "space between the sky and the earth" [closed]

Troposphere

the lowest region of the atmosphere, extending from the earth's surface to a height of about 6–10 km (the lower boundary of the stratosphere)

[Lexico]

but that might be a bit "sciencey" sounding for your suggested sentences.

and interpretative suggestion might be something like:

Blue yonder

or similar

The word "azure" is a term that is suitable, although it is literary.

(SOED) azure 5 The unclouded vault of the sky.

(SOED) vault 1 Archit. A continuous arch, or a series of arches radiating from a central point or line, used to form a roof over a space in the interior of a building; an arched roof or ceiling. ME. b An arching structure or covering resembling a vault; esp. (more fully heavenly vault, vault of heaven*) that formed by the sky (chiefly poet.) LME.

Since clouds are well known to have volume, as well as being found over a range of altitudes that can be quite high and quite low, the term "azure" does refer to a space that occupies a fairly important part of what lies between the elusive boundary of blue expanse above our heads and the earth.

(ref., 1995) Shortly after the third jet had disappeared into the azure, a mortar shell landed on the far side of the airport, well away from any aircraft and apparently causing no damage.

• She lost sight of the swans that vanished into the azure.

Personally I think this is a concept that isn't recognized in English. For me, the sky is anywhere I don't see ground or something attached to the ground. I would consider a bird flying a few inches above my head to be "in the sky" (assuming we weren't in some kind of structure). The idea of empty space between the sky and the ground is nonsensical to me. No matter which word or phrase you use, I don't think you will be understood.

You could try distance.

A stretch of space without designation of limit; an expanse.

a land of few hills and great distances.

[American Heritage Dictionary]

And therefore you could write your sentence thus:

She lost sight of the swans that vanished into the distance.