Do the denizens of The Pits have personal "souls" of their own?
They do, after all "Each living creature has a soul and this soul occupies the body. When a creature dies, the soul leaves the body."
For example, a Succubus, Balor, or any of the other commonly known demons/devils?
Well, the demons and devils distinction comes from D&D, we no longer have that.
If their body is destroyed on Layonara do they have the same death experience as a mortal which is described in the new LORE?
Is there any difference if their body is destroyed on Layonara or their own native Pit?
If they die in their Pit (or if somebody from Layonara proper dies in a Pit), their Soul would still go to the Plane of the Lost first.