How can elements be selected and moved properly? I'm having trouble selecting and moving elements in a model.
The recording and file is here.
Steps to replicate:

Select element (i.e. face)
Move outward
Move inward
Tap on first selected element

Expected:

When moving outward, faces should not fold.
When moving inward, form crater.
When the former face is tapped, it should become selected.

Results:

When moving outward, faces fold.
When moving inward, the topology does not dip below the original surface.
When the former face is tapped again after moving inward, there is no feedback.

I think I might have accidentally ripped everything. How can this be fixed?
The problem is with the mirror modifier. You already have your entire object completely meshed out, but are then mirroring it over on itself. The extra geometry that you are unable to edit is actually the mirrored mesh from the other half.

You should delete half of the mesh so that the mirror modifier isn't flipping the mesh over onto itself.