How to hang stuff on wood paneling
How thick is that paneling likely to be?


No one here can make that assessment for you; you will need to inspect the wall, by either drilling a hole at the intended mounting site, or opening the wall up.


  Is that same double-thickness situation (which looks like plaster over drywall, or maybe two layers of plaster) likely to be under the paneling?


Again, you will need to inspect the wall you intend to hang the shelf from and not make any assumptions of continuity of construction. Your inspector spoke out of turn here; unless he was the original home owner or carpenter that put the paneling up.


  If not, what is?


It's a 70 year old house - there is no reason to assume that it has never been remodeled in the past, and that any old work dug into, is original to the house. What's behind door number 2? Look.


  Am I going to need any special kind of hardware to hang shelves on wood over plaster over drywall over a stud?


Probably not - as long as the paneling has substantial thickness, and is properly affixed to the underlying framing. Neither of which you know at this point, and no one here can tell you, unless they've had that wall open themselves.

In earlier period paneling jobs, is wasn't unusual for the carpenter to panel a wall, not unlike they would panel a floor. By furring the surface in an apposing direction or at an angle to the framing first, then the finished panels at whatever angle was desired - without plaster or drywall at all.