You have some scrambled terminology in your question. First, is the magnet permanent? If it is only a soft iron core that becomes a magnet when the coil is energized, it is an electromagnet and can operate on either AC or DC. For AC operation, put one or more copper slugs or shorted single turns of heavy gauge wire that will hold the field while the power reverses direction.
Second, actually no it isn’t storing current except in the slugs in the AC version unless the coil is shorted. The energy is actually stored in the H field (magnetic field). When it collapses it induces a voltage. If there is a closed circuit the voltage will cause a current to flow.
Third, since the coil is the inductor, question 3A makes no sense. Connect power to the coil; it will create the field which will react with your magnet causing it to move. If it is a permanent magnet, the polarity of the power will determine if the magnet is sucked into the coil or if it will shoot out of it. A soft iron bar will always be sucked into the coil in both the AC and DC versions.
Last, Get some 20 AWG varnished copper wire, the tube from a toilet paper roll and a piece of iron that is a little smaller than the tube so it moves through it smoothly. Wrap the wire around the tube, set the iron bar about half way in and connect a battery. Next, bring up a magnet and let it stick to the core and note the differences in behavior with a magnet instead of just an iron core. After that, you can go crazy with your experimentation.