1) It's a button or coin cell [1] (halfway down). So the conductor is the steel casing, the e- travel as you put a wire from either side wrapping around the cell. The copper current collector is only on one side. You're right in that they don't really fully describe the cathode in the methods portion. They simply state the cathode consists of a redox center (S or MnO2) embedded in electrolyte and carbon in contact with the copper, which is pressed against the glass electrolyte. It could be a slurry that they press, or something that has water, but that is never mentioned. I think the problem is that nobody really knows what the atoms at the cathode are doing. I would be hesitant to say anything is aqueous, but I guess there is the possibility.
2) Cathode is reported at 0.06 mm. They show SEM pictures of Li plated on the cathode, and directly call the cell a metal plating cell. From their conclusion, "With the Li-glass and Na-glass electrolytes, we have demonstrated in this paper one possible new strategy in which the cathode consists of plating the anode alkali-metal on a copper–carbon cathode current collector at a voltage V > 3.0 V." Additionally, from the results, they "examined the electrodes with the naked eye and with SEM EDS analysis, as shown in Fig. 2, which indeed shows lithium plated on the cathode current collector and no evidence of metallic lithium remaining on the stainless steel at the anode or the anode side of the electrolyte after full discharge of the lithium anode."
3) The e- are pulled off by the voltage applied during charging with any voltage higher than the work function of whatever state the plated Li is in (I think), causing a charge imbalance and Li ionization. This is balanced by the ionized Li moving back through the electrolyte to the anode side. The role of the sulfur is debated and I would say not understood, but they claim it's a redox center, where redox does not actually happen. Here is the key part:
We therefore conclude that the sulfur acts as a redox center determining the voltage of the cell at which electrons from the anode reduce the Li+ at the electrolyte/cathode interface to plate lithium rather than reducing the sulfur, so long as the voltage remains above 2.34 V; below 2.34 V, the S8 molecules are reduced to Li2Sx (1 r<= x <= 8) and the lithium on the anode becomes exhausted after 28 days in the cell of Fig. 1. The cell reaction was no longer reversible after this full discharge. At voltages V > 2.34 V, the cell is rechargeable and the sulfur is not reduced. The Fermi level of the lithium plated on the carbon–copper composite cathode current collector is determined by the Fermi level of the cathode current collector, whereas the Fermi level of the lithium anode remains that of metallic lithium, but the cell voltage is determined by the energy of the redox couple of the unreduced redox center
That is the part that I think is confusing a lot of people, myself included, on how that happens exactly. They have some evidence that supports their claim, but I'm sure that's the next paper, how this actually works.
4) With a cathode that is 60 microns thick that they dry from a slurry, there is no real basis for "apertures" through the thickness, especially in a disordered glassy structure. Instead, ion transport is through vacancy mediated diffusion. This does have a dependence on atomic radii, but from what I'm reading, is more dependent on the atom species being ionic as opposed to neutral. Again though, the important portion of the finding is that the electrolyte is solid, not liquid, and therefore during lithium deposition on either the cathode or anode, ions cannot easily preferentially grow dendritic structures as opposed to evenly plating the anode/cathode.
I would agree that more characterization needs to be done of the role o the components during all stages of charging and usage. Comparing a recharged and emptied batteries in a microscope would be helpful, as well as more careful chemical analysis as to what is actually going on.
[1] http://batteryuniversity.com/learn/article/types_of_battery_...