There is certainly an issue with the ethernet port which I will summarise here:
1) can ping over wireless but cannot download anything
2) cannot ping over homeplug devices, even though the Owon is connected straight to the router/hub
3) can ping and download only when the software is running on a PC which is directly connected to the LAN, ie no wireless and no homeplug
Number (2) maybe be a hint: when I connect my PC to the LAN through the homeplug device (the system using the electrical cables), it cannot ping or "see" the Owon - but just the Owon. I can ping and connect and use everything else in the house, including other PCs, TVs, Blue-Ray players, routers, wireless access points, printers, everything.
Observing how the oscilloscope software works, I have made another assumption : it may be that in the handshake/start/stop download, they have used simple "waits" rather than proper synchronisation protocols, and these "waits" work only when you are on a real ethernet cable with specific latency. But when you are over the wireless / homeplug, there is added latency that probably throws all those primitive "waits" off. Just an idea.