The DC means it has an electromagnet and a trip switch to make the bell clang, so that magnet depends on current running one way. Try to run it without an inverter and the magnet will just set there and hum and not have any direction
In principle, yes, but if the frequency is too high the mechanical inertia would dampen the oscillations and the armature would not swing to strike the bells.
Why Not. Absoutly yes.
No