Solid reasons for switching off mobile devices on air planes
Are there reported cases of mobile devices influencing airborne safety (with or without radio on)?


Hearsay is hearsay. The plural of anecdotes is not data. There are plenty of anecdotes, but none that have been reproduced under controlled conditions.


Can an operating devices in "airplane mode" even hypothetically influence an aircraft?


This depends on how you define "aircraft". Can, under laboratory conditions, radio-emitting devices (e.g. devices not in "airplane mode") cause momentary issues with the aircraft avionics that use RF (such as GPS and radio-location beacons)? Yes.   

Can such interference affect anything other then the displayed information in the cockpit? No.
In other words, the interference won't crash the plane, cause the control surfaces to deflect, or anything. It just makes the position/heading/whatever readout display incorrect information.

The Wikipedia page on this is a good source. There are already a number of airlines that are allowing people to make cellular calls during flights, let alone use devices in "airplane mode".