LDPE > Move Adjacent Data

[Nils Schmidt]

sometimes it turns itself off with no reason.

By default it turns itself off when pasting new stuff. But you can control this (tools->options->misc options)

"Move Adjacent Data" will deactivate when you paste something into the model. This is a protection, because the pasted objects are adjacent to their original selection (to be precise: they are on the same place!) thus I expect that you don't want to move the original and the pasted copy.

I tried moving vertices in the 3D editor, but I can't. Neither with or without the button on. Does it need to be a LPE VERTICE or can it be any?

I don't know how/why this happens to you, use it routinely to do exactly the things you tried and don't work for you. The only thing it doesn't do (of course) is to move primitive or subpart elements in the selection, but it does cheerfully moves anything non-primitive in selection (and detach them from primitives if needed).
[...]
Can be any. But of course if the vertex is carried only by a prim/subpart it needs to be copy/pasted to a lpe vertex first (and move adjacent data re-enabled if needed)

Exactly!