Lenz’s law states that the polarity of induced emf is such that, it tends to produce a current which opposes the change in magnetic flux that produced it.
Yes, emf will be induced in the rod as there is change in magnetic flux.
When a metallic rod held horizontally along East-West direction, is allowed to fall freely under gravity, i.e. fall from North to South direction, the intensity of magnetic lines of the earths magnetic field changes through it, i.e. the magnetic flux changes and hence the induced emf in it. When we increase the number of turns, the induced emf will increase because induced emf is directly proportional to the number of turns.