Does storing apples alongside potatoes keep potatoes from sprouting?
Ethylene has been used in storing potatoes for a while, see this article. In it, they address the puzzle of a ripening agent acting to prevent sprouting:


  Sprout suppressant properties   of   apple   ‘vapours’   were   first   reported   in   1932   and   subsequently,  ethylene  and  apple  ‘vapours’  in  1933  (Elmer  and  Huelin,  respectively).    This  was  preceeded,  however,  by  work  demonstrating  a  ‘dormancy  breaking’  effect  of  ethylene  (Rosa,  1925).    These  apparently  conflicting  responses  to  ethylene  were  resolved  with  the  publication of work showing both effects, but under different ethylene exposure conditions.


The mechanism of the different effects are largely unknown. The cited article (by the British Potato Council) gives some references. Potatoes treated with ethylene do have slightly different properties (they mention the color after deep frying), and there is ongoing research to get an overview of all the differences.


  [OP:] Chemically speaking, do we actually know the exact reason and the necessary conditions to make this work?


We don't know the exact reason, but there are companies who figured out how to make this work in a warehouse under controlled conditions. I think on your shelf would be more difficult because you have no way of monitoring the ethylene concentration, so the apples might influence the potatoes in etiher way.