Thanks for reading. Good to hear that you find the technology exciting. The battery's technical details obviously escape me, but I did a little digging to find some answers to your questions. The 16-inch prototype batteries only contain about 1200 Wh, but that could change as they approach commercialization. For now, the batteries are small enough that the plan is to stack them into shipping containers for a combined 2MWh storage capacity: http://www.ambri.com/storage/documents/ambri_brochure_web.pdf.
As for keeping the battery interior melted: "The whole device is kept at a high temperature, around 700 degrees Celsius, so that the layers remain molten. In the small devices being tested in the lab, maintaining this temperature requires an outside heater, but Sadoway says that in the full-scale version, the electrical current being pumped into, or out of, the battery will be sufficient to maintain that temperature without any outside heat source." (http://web.mit.edu/newsoffice/2009/liquid-battery.html).
Hope this helps.