After I use up all the materials in a mine or quarry, what happens to it? Does it just remain as an unusable chunk of land, or does it slowly recharge or decay, or something else entirely?
The buildings used by a mine and quarry can be removed, but the land they used will not be usable again. This is covered in the Help Menu of the game. The relevant sections are below.
Mines cannot be fully removed. The building can be taken down, but the area that the mine takes up won't ever be available for use for other purposes.
Quarries cannot be removed. After the quarry is no longer useful, the building can be taken down, but the hole that has been dug in the quarry won't ever be available for any other use.