Does the ring (in the picture) experience Lorentz force? Does the ring in the picture below experience Lorentz force? Or should I say it's just some magnetic force? (Notice that the poles repel eachother.) I am unsure because of the setting with the electromagnet.

 A: 
Does the ring in the picture below experience Lorentz force?

Rather yes than no, but we need to make clear in what sense.
The Lorentz force is a somewhat confusing term, because people use it to name different concepts.
The term has some history, but nowadays it most often refers to a force acting on a tiny charged particle in an external electromagnetic field.  Depending on the source and context, it means either the magnetic force $q\mathbf v\times\mathbf B_{ext}$, or the entire electromagnetic force $q\mathbf E_{ext} + q\mathbf v\times\mathbf B_{ext}$, where $\mathbf E_{ext},\mathbf B_{ext}$ are the external fields at the point where the particle is.
For example, think of the electrons circling in magnetic field of a cyclotron, or electrons flying through a CRT tube (old-generation kind of TV). The force electron experiences there is accurately given by the  Lorentz force formula, where the external fields are determined by the magnets of the cyclotron and the voltage imposed on the metallic plates of the CRT.
However, the ring in question is not a tiny charged particle. It is rather big in the sense that the values of electric and magnetic field vary across the ring. The Lorentz force formula in the above sense is not applicable for the ring as a whole.
The total EM force in such cases is calculated in a different way. The standard approach is based on the formula
$$
\mathbf F = \int_V \rho\mathbf E + \mathbf j\times \mathbf B \,dV
$$
which, unfortunately, is sometimes referred to as the Lorentz force as well.
This formula is quite different from the first one. An integration over region of space $V$ containing the entire body is involved. Second, although the integrand $\rho\mathbf E + \mathbf j\times \mathbf B$ is quite similar to the expression $q\mathbf E_{ext} + q\mathbf v\times\mathbf B_{ext}$, there is an important difference; the first formula (for tiny particles) involves velocity of the particle $\mathbf v$ and external fields, but the expression in the integrand here involves total current density $\mathbf j$ in the body and total electric and magnetic fields.

Or should I say it's just some magnetic force? 

The process illustrated in the picture involves electric and magnetic field changing in time inside the ring. In such cases there is no reason to think the force is purely electric or purely magnetic, without careful calculation. I'd describe it as electromagnetic force due to solenoid acting on the ring.
A: No. Lorentz force act on moving charges in a magnetic field. The current in your ring is created by a changing magnetic field (induction). Lorentz force also acts perpendicular to the particle velocity vector and the magnetic field, which is not the case here.
A: First at all think about induction processes.


*

*An electric current in a coil induces a magnetic field. This happens because the electrons in the coil get accelerated in a circle and any acceleration (the motion in a circle is an acceleration) of electrons induces a magnetic field.

*A magnetic field induces a electric field in a moving conductor. Because the aluminum ring is a conductor there will be a electric current in the ring. Than faster the ring is falling than faster the electric current will increase.

*Last step, an changing electric current induces an magnetic field. For all this induction processes one can use the hand rules and thumb rules, you find them on Wikipedia. All this rules show, that the induction of both the electric and the magnetic field from the vis-a-vis field are directed perpendicular to each other. You will have used the rules right, when as a result the magnetic field from the coil and the magnetic field from the aluminum ring will be directed counterwise.
Because of the gravitational acceleration of the ring, the delayed reaction of the induction processes and the energy transmission from the electric current from the coil to the ring the ringt gets accelerated in a "explosive" way.
The Lorentz force is the deflection of a current-carrying conductor in a - not-parallel to the current - magnetic field. If lower a wire to the coil there will be an induced current and the coils magnetic field will deflect the wire or to or away from the coils symmetry axis. So if one let the ring fell not properly symmetric to the coil, the ring get deflected additional sideways. So sometimes the experiment does not give the expected result. Try it.
Edit: If I remember right the ring has to be from aluminum due to the property of aluminum that there will be eddy currents which are the cause of the delay. Hope to get some comments to get enlightened.
