When I attach the Wifi Extension to the top of my master brick, it shows up in Brick Viewer but the FW Version is stuck on "Querying..." and the Master Brick tab reads "Waiting for WIFI Extension 2.0 firmware version...)

In Updates / Flashing, the extension tab lets me select "WIFI EXtension 2.0 (0.0.0) @ Master Brick 3.1 [UID] (2.5.2)", and I can select firmware and hit Flash. Doing so pauses on "Connecting to bootloader" for about a minute before giving me "Error during Extension flashing: Failed to connect to ESP8266". I have tried multiple different cables and multiple master bricks, and all give the same problem. Is it just a bad extension module?

Which Wifi Extension do you have? There is version 1.0 and 2.0. 

I guess you will have version 2.0. Is it possible that the Master Brick and the Extension are not stacked properly? It sounds that there are connection problems.