No, moving a ferrite core through a coil won't generate a voltage. You would need to move a magnet to generate a voltage.

Look for a vibration switch like one of these. If you want more control, you could use an accelerometer and a microcontroller to trigger it from a specific amount of movement.

I haven't done a course in electromagnetism yet, but as far as I understand, the ferrite core is just a piece of metal with no magnetic field, so moving it doesn't induce a voltage

I think it would, however, change the inductance, just like the iron core in a transformer does

If your coil was oscillating, then perhaps an iron core moving through it would cause perturbations which are detectable. But that would require extra logic to compare the expected oscillation frequency with what the coil is actually oscillating at.

Since you say that tilt switches are not an option -- for reasons I'm not entirely sure I understand -- another option is to have a linear Hall effect sensor mounted nearby a small magnet. If the magnet moves relative to the sensor, then that is a change which can be acted upon. A linear sensor makes it possible to use a trim pot to tune the sensitivity.

No, that won't work.

A vibration switch will work.

If that's not sensitive enough, another option is using a piezo element coupled to the case to detect vibration, with an op-amp or hex inverter to buffer + trigger the 555. However if you couple it too closely with e.g. the floor or furniture it will pick up nearby footsteps or cars. Might be good depending on the situation.