1. I assume the galvanometer has a very fast response time in the above simulation.
If the response is much larger than the period you move the magnet (in and out),
then the galvanometer will only show small amplitude.
The effect due to finit response time was not consider/implemented in the above simulation.
2. There is always induced current if the solenoid form a close loop (to drive the galvanometer).
Even ther is a current shown in the simulation, it was designed to show the direction of the induced voltage.
The induced current is assume to be very small to cause any significant magnetic flux change.
3. The energy produced is coming from work done by external force.
The magnet will not be demagnetize.
The magnet might be demagnetize very small amount due to your hand warm up it's temperature (normally the effect is much smaller than the experiment error).