4 speed auto in a w124 is purely mechanical/hydraulic i.e. no provision for star (leccy diagnostics)
I'm confused by the wording the in the OP... 'couldn't start the car' and 'tapping starter motor' fits with inhibitor switch BUT 'in reverse or any other gear it just seems to roll'
When i initially read the OP i was thinking engine running but no drive in any gear? Are we infact saying you couldn't start the engine but tried pushing it in drive/reverse and it rolled as though it was in neutral?
The can't select park thing... can't take the key out of the ignition in a w124
unless it's in park so presumably it was in park when you left it in the morning? There's yet another safety device that prevents you from shifting out of park with the ign on unless the foot brake is applied soooo i'm thinking the gear linkage as unimoged... you can move the gear lever in the car about because it's no longer connected to the actual gearbox. That'd explain why you can't get back into park (the gearbox is still in park) but not the 'seems to roll' part of the OP
The key/foot brake safety interlock for park is done by cables, pics here...
shifter stuck in park - Mercedes-Benz Forum
As said the inhibitor switch prevents the starter from working when it's gear but it will start in both neutral and park