How "full-motion" is a full-motion simulator?
Full Motion is full in the sense that it can generate cues in all 6 Degrees Of Freedom: fwd/aft, left/right, up/down, pitch, roll, yaw. The term came in use to distinguish between 6-DOF systems and the earlier 3-DOF systems which only produced pitch, roll, heave cues.


  Do they just simulate the major motions of the airplane in response to the flight controls...


They simulate the aircraft response by stimulating the sensations felt by the inner ear, making use of its limitations as also mentioned in this answer.



The two side-questions about reproduced vibrations boil down to the frequency response of the motion system: the ability of the system to reproduce commanded vibration signals, as measured at the upper platform with the payload installed. Older hydraulic systems, such as depicted above, had a very limited frequency response - the servo valve opens and oil flows through, creating dynamic flow effects and springiness from the compressibility of the column of oil supporting the payload. Yes, oil is compressible. Maximum perceptible cues up to 5 Hz.



Electric systems have an electric motor at the bottom of the actuator. Electric field creation happens almost instantaneously when commanded and the connection to the upper platform is through the metal screwjacks inside the tubes: frequency response is almost an order of magnitude better than that of hydraulic systems. Maximum perceptible cues up to 30 Hz. Enough for engine vibration reproduction, severity depends on the magnitude of the amplitude.

The issues with reproducing severe vibrations at the motion upper platform are:


The cabin construction must be able to rigidly stay in one place and support the visual projectors without starting to wobble the image that the pilots are looking at - the rigidity of the upper construction must exceed the frequency response of the motion system. We've seen simulator cabins bobbing about like a gelatine pudding when subjected to serious amplitude/frequency combinations. 
All instruments, electronics, cabling etc. are subjected to the alarming vibrations as well. 


Both issues above are not an issue if the constructional design of the cabin etc. has incorporated the vibration inputs. The alarming vibrations are extremely effective training cues, and can be faithfully reproduced by an electric motion system plus suitable cabin construction.

Another way of producing severe high frequency vibrations are by means of a seat shaker or cabin shaker, used in the olden days of hydraulic motion, particularly with helicopter simulators for reproducing the main rotor vibrations. The seats would be mounted on up/down platforms with a stroke of several centimetres, actuated by a separate hydraulic small actuator.