Moving objects out of a object layer deletes the object layer?

Maybe what you think was a layer, actually was a group.

Was the object layer deleted or just moved elsewhere?

Things can seem unclear when it comes to Illustrator's layer and sublayer relationships. The primary layers you create (and the one that is present by default in a new file), are indeed layers in every sense. The 'sublayers' that appear when objects are addeed to primary layers (in your post, you refer to them as 'Object Layers') are, in effect, the objects themselves. Delete the object, and it's gone from the stack; the 'object layer' was the object, so with the object gone, nothing (no empty layer) is left behind.

As Ton explained, for a sublayer to have permanence, you must expressly create it. (And subsequently, objects created on that layer—or moved into it—become object-sublayers of your created sublayer.)

Without seeing, I think that you are moving objects, which show as separate items on a layer, but they are not layers.

When moving the object, the item name is moved as well.

You need to create a sublayer yourself if you want to keep that structure after moving objects.