1) The cross section is given in an image in the original article, and explained in a Medium post [1]. It is Li-(fancy) glass-(Cu/C/S combination) encased in steel. I think for now, you can ignore the steel. Nothing is aqueous, as it's a solid state battery.
2) During discharge, e- leave the anode, travel through the wire and end up in the Cu/C/S cathode. This leaves a Li+ cation which then travels through the glass, (most likely) does not remain there, and deposits itself on the cathode. Hence, in the original paper [2], they see a build up of lithium on the cathode via SEM. From what I understand, the contentious part of the paper is that the researchers state that the S provides the driving force for the reaction, but then isn't reacted (the main point in [1]), hence the "perpetual motion" attack. The deposited lithium does have a sort of screening effect on the cell voltage, slowly decreasing it with discharge time. From my understanding, the battery capacity is limited by the amount of Li before it is all oxidized, and not dependent on the amount of reduction the cathode can handle, very interesting. This cross section at full discharge would be nothing-glass-Li-(Cu/C/S cathode), again encased in steel, but ignore that.
3) During recharge, the opposite happens, e- are forced back to the anode, and the preponderance of e- cause a driving force to remove the Li that has been deposited on the (Cu/C/S) and transfer it back to the metallic Li side. The key part is that the electrolyte shouldn't be holding onto any ions, if the paper description is to believed. The Medium article states that some sort of reaction may be occurring in the electrolyte, I can't really weigh on on that. Once all of the Li is back on its anode side, recharging is done.
4) It is important that the glass electrolyte is a solid, not that the cathodic atoms/ions cannot enter it. They most likely can, it would just take a recharging voltage much higher than the one they use to recharge the cell. The fact that it is a solid means that lithium dendrites (tree like growths) cannot grow during redeposition, as dendrites that span the distance between anode and cathode short circuit the battery.
Feel free to ask me any questions about it, I am not a battery person so there's a high chance I got something wrong, but I am a MSE grad student so I might be able to say something that sounds intelligent.
[1] https://medium.com/@steingart/a-potential-big-deal-in-batter...
[2] http://pubs.rsc.org/en/content/articlepdf/2017/ee/c6ee02888h