The magnet has an unchanging magnetic field: very strong near the magnet, and weak far from the magnet. How did the magnetic field through the coil change as the magnet fell toward it? How did the magnetic flux through the coil change as the magnet fell toward it?

The magnetic field through the coil at first increases steadily up to its maximum value, and then decreases gradually to its minimum value.

Explanation:

At first, the magnet fall towards the coils;  inducing a gradually increasing magnetic field through the coil as it falls into the coil. At the instance when half the magnet coincides with the coil, the magnetic field magnitude on the coil is at its maximum value. When the magnet falls pass the coil towards the floor, the magnetic field then starts to decrease gradually from a strong magnitude to a weak magnitude.

This action creates a changing magnetic flux around the coil. The result is that an induced current is induced in the coil, and the induced current in the coil will flow in such a way as to oppose the action of the falling magnet. This is based on lenz law that states that the induced current acts in such a way as to oppose the motion or the action that produces it.