So to start, yes, E is ultimately the kinetic energy, at least in these cases as we're talking plasma.
And as to calculating 𝜎, while we can't do it in the lab, we have a sky full of stars to look at. We can make good estimates of the mass of a star, so that gives us the amount of energy that must be generated for it to be stable, and from that we can determine the required reaction rate. We can also make good estimates of the core temperature, so that gives us the rate for a given E. Do that for lots of stars and you get a curve. So it can still be experimentally determined, just not in the lab.
As to the second part of your question, the S-factor is just a re-writing of the cross section in sort of the same way that we use the reduced Planck constant to save you from writing all those 2pis. In this case, the idea is to bring out some things that are hidden in the numbers if you just consider cross section.
The key issue in fusion is that the rate is dependent on two factors, one is the kinetic energy in the center-of-mass coordinates which is being used to overcome the Coulomb barrier, and the other is quantum tunnelling. The former is essentially classical, if the energy is over X then it can fuse, otherwise it can't. The second is essentially geometrical, but based on the de Broglie wavelength, which also has an energy dependence.
When you look at the S-factor formula you posted, you can see these terms. The 𝜎 is accounting (roughly) for the classical side, and the 𝑒2𝜋𝜂/E is the tunnelling side - the top part is the Gamow factor, and the 1/E is the de Broglie wavelength.
So what that leaves in S are the nuclear bits. Just because two protons have enough energy and/or tunnelled doesn't mean they will react, there's all sorts of things going on inside. So we isolate all of that in S.
When you look at a graph of cross section vs. energy you get something that is an exponential curve. As you approach the Coulomb barrier energy the rate goes up and up and beyond that it flattens out. But there's very interesting things going on in that curve that you can't see because it's exponential.
So if you look at the exact same reaction using the S-factor instead, you see something very different. For higher energies, the curve is flat, but at energies well below the Coulomb barrier there's all these peaks that were formerly invisible.
There's a series of YT videos that covers this, you can start with this one. Around the 11 to 13 minute mark he explains why we have S factor. On the figures, you'll see that the cross section has this big exponential drop-off at lower E, whereas when you convert that to S-factor you get something that is pretty much linear for higher energy and has interesting peaks at lower energy. This gives you more insight into what is "going on in there".
As to the "set to zero" part, I can't say, because S(0) is zero. Perhaps they are setting one part of the equation to zero or the other, like maybe setting the tunnelling term to zero to reveal the classic cross section?
