Conservation of energy in Lenz's law

You can actually arrange a simpler experiment for this. Suppose that Lenz's law were reversed and induced currents reinforced the change of the magnetic flux. Now take a single loop of wire, and suppose that we produce a small current in it (with a battery or a magnet - it doesn't matter). An increasing magnetic flux is then created through the loop.

This changing flux will now induce a current in the wire loop. You'll see by the right hand rule and the modified Lenz's law that the induced current goes in the same direction as the existing current. So the induced current reinforces the existing current - the total current increases and so does the magnetic flux, which induces a further increase in the current etc... With a simple loop of wire you could power a city.

In the actual case the induced current resists the increase in mangetic flux and opposes the current already present in the wire, slowing and eventually stopping the growth in the current. This is the operating principle of an inductor.

The difference between the solenoid and the magnet is that the magnetic field in the soleniod is a transitory thing, created by our motion of the second magnet. In the first case, the fixed magnet has it's own magnetic field and the system already has energy in it - put there when we first moved the magnets apart (or created the second magnet). We can get that energy back as work done on the magnets as we bring them together.

In the second case, there is no magnetic field in the solenoid until the approaching magnet starts to move - and hence no energy in the system. It is the work we do to move the magnet that creates the field, and if that field then acted to accelerate the magnet then obviously we would get out more energy than we have put in, and there is your violation of conservation of energy.

Lenz law actually is that induced current always tends to oppose the cause which produce it.So in order to do work against opposing force we have to put extra effort. This extra work leads to periodic change in magnetic flux hence more current is induced. Thus the extra effort is just transformed into electrical energy which is law of conservation of energy.