1) With my current cars (spanning 1988-2006, both manual transmission and automatics), one can can turn off the ignition with the keys while moving, however to remove the keys they must be turned to the lock position (the automatics also must be in park but the MTs don't care). Steering and braking forces go way up with the engine off and the airbags deactivate - wise people don't do this. With my earlier cars, (1950 & 1960), the keys could be pulled (after turning off the ignition) at any time - no transmission or steering interlocks.
2) Good question about whether our son's car, a 2016 Mustang, is defective and maybe he should check with the dealer. At the end of the section on starting/stopping the engine the owners' manual addresses the dangers of shutting off the ignition while the car is in motion, but for those who might do it goes on to advise what to do until and after and after the comes to rest. The manual is little ambiguous, it doesn't specifically instruct how to shut it off while moving but in addressing it implies that it can be done.