Are the field lines around a magnet (and the repulsion felt between magnets) EM radiation? If not, is it those field lines which turn into EM radiation when a charge is accelerated?
 A: Taking your questions in order:

Are the field lines around a magnet (and the repulsion felt between magnets) EM radiation?

No.
It's important to note that while field lines are a helpful visualization tool, they should not be confused for the fields themselves.  The electric and the magnetic field exist at all points in space, even the points in space that a field line doesn't pass through.
But you can certainly ask "is the magnetic field created by a magnet an example of EM radiation?"  And the answer to this is "no", because for a configuration of electric and magnetic fields to qualify as "radiation", they have to be carrying energy arbitrarily far away from the charges & currents that created it.  Roughly speaking, for this energy flow to happen, you have to have both an electric field and a magnetic field present, and their strength has to fall off (no faster than) proportionally to $1/r$, where $r$ is the distance from the charges & currents.  For a bar magnet sitting in space, there is no electric field, and the magnetic field is proportional to $1/r^3$;  so there is no energy transport, and no EM radiation.

If not, is it those field lines which turn into EM radiation when a charge is accelerated?

More or less (though again, you should be thinking of this in terms of the fields, not in terms of field lines).  Again roughly speaking, an accelerating charge creates an electric field that changes in time.  By Ampere's Law, if there's a changing electric field, there must also be a magnetic field;  and by Faraday's Law, this changing magnetic field creates a changing electric field;  and so forth and so on.  It can be shown (via some complicated mathematics) that this self-reinforcing cycle leads to an electric field and a magnetic field proportional to $1/r$.  Effectively, the changes in the field lines when you accelerate a charge create additional fields (both electric and magnetic) that have the right properties to carry energy away from the charge.
A: No, because it is a static situation.
An accelerated charge generates EM radiation and if it is strong enough, could make iron filings on a piece of paper oscillate. So yes, theoretically, moving field lines could show EM radiation.
We can make an analogy with gravitational field and how strong is the force downward depending on the steepness of points in the dunes of a desert.
If instead of sand there is water in the sea, the outcome is waves, not static dunes. And the water waves are indeed called gravity waves.
