The "lines" you see when viewing iron filings around a magnet have more to do with the fact that they are tiny slivers of iron, and less to do with magnetic field lines as one normally talks about them.
Also, over the length scale of one of these slivers, the magnetic field is largely constant, and a ferromagnet (or magnetic dipole) placed in a constant magnetic field will not accelerate (it will, however, align itself with the field). Once two slivers line themselves up head to tail, the field they create around them makes it more favorable for other slivers to join the chain rather than to lie haphazard, because the filings distort the field around them. So it is simply energetically preferred for these slivers to line up head to tail and form longer chains, but if you look closely, the chains break and merge.
Magnetic field lines are just a way of visualizing magnetic fields, in the same way that electric field lines are used to visualize electric fields (lines of force). There are no "gaps" between true magnetic field lines -- they occupy all space. We just draw them that way to convey a sense of their intensity.
I also don't quite agree with the statement that friction prevents them from clustering on the magnet. It's a bit more complicated than that, and, indeed, you can watch the same behaviour in air by suspending a magnet above the filings and allowing them to lift up. Once the filings start attaching themselves to the magnet, a magnetic circuit is created which changes how the field looks.
One of Maxwell's equations is
$$ \nabla \times \vec{E} = - \frac{d\vec{B}}{dt} \, .$$
Consider an imaginary disk whose normal vector is parallel to the axis of the coil and which is inside the coil.
If you integrate this equation over the area of that disk you get $^{[a]}$
$$\mathcal{E} = - \dot{\Phi}$$
where $\Phi$ is the flux threading the disk and $\mathcal{E}$ is the EMF drop around the loop.
This is called Faraday's law.
So, for each imaginary disk inside your coil we get some EMF as the flux through that disk changes in time.
Now think about the bar magnet's descent.
Suppose we drop it starting way above the entrance to the coil.
It's far away, so there's no flux and no EMF.
As it descends and gets close to the entrance of the coil, some of the magnetic field from the magnet threads the top few imaginary disks of the coil.
The time changing flux induces some EMF.
This is the initial rise in the red part of the diagram.
As the magnet continues to fall, and enters the coil, more of its magnetic field is threading imaginary disks in the coil, so as it moves the time rate of change of total flux increases, so the EMF goes up.
Note that the field lines above and below the bar magnet point in the same direction.
At some point, the bar reaches the middle of the coil.
At this point, the amount of flux added to the top half of the coil by a small motion of the magnet is equal to the amount of flux removed from the bottom half.
Therefore, at this point the EMF is zero.
This is the midopint of the diagram where the EMF crosses the horizontal axis.
The falling part of the red section is just the approach to the mid section of the coil.
As the bar magnet exits the coil, more flux is leaving than is entering, so the EMF versus time in the blue section is just the opposite (except for the stretching which you already understand) of the red section.
[a]: On the right hand side, the area integral of the magnetic field is the flux $\Phi$ by definition, and the time derivative just goes along for the ride.
On the left hand side you are doing an area integral of a curl of a vector, which by Stokes's theorem is equivalent to the line integral of the vector itself around the boundary of the area.
The line integral of the electric field vector is the EMF by definition.
Best Answer
Electricity and Magnetism are the two sides of the same coin. Magnetism is produced only due to the effect of moving charges, every charged body in motion acts like a magnet. Therefore, in the atoms of a magnet, the electrons moving around their nucleus are actually magnet! Now one would ask if it was so, then all the elements around us should act like magnet because all of them have electrons. But, the answer is - NO! It is because even if an atom as a whole is magnetic but they all are aligned in such a manner that their net magnetic effect is cancelled out and the element is non magnetic.
In certain elements, the atoms are aligned in such a manner that the magnetic effect do not cancel out. Such elements are known as - "Ferromagnetic Elements" Therefore the magnetic field of a bar magnet is not because of its pole strength but actually because of its perfect alignment of magnetic atoms and that the atoms are magnetic because of the negatively charged moving electrons.