Streaming my Xbox One to PC won't work while connected with an ethernet cable to the same...

So as long as the Xbox and the PC are connected to the same local network (your house's internet), you can stream the Xbox to your Windows 10 PC.

For the best connection with the LEAST amount of input lag, both the Xbox and PC should ideally be connected to the router via ethernet cable.

You want force Ethernet connection for xbox or pc? If you connect Ethernet cable to Xbox, and evertyhing work normally (dhcp server give ip address, etc) xbox one ALWAYS use Ethernet cable connection. I think, your powerline adapters NOT configured right,
do you remember pairing powerlineadapters? Can you test Ethernet connection? Maybe you can use laptop or something another pc to test Ethernet connection connectivity... Mayby your pc is configured static ip? And pc working well when you use same Ethernet
cable, but when you connect Cable to xbox you cannot connect to Internet, because your Xbox want use dhcp.... Can you move xbox temporarily to same room where you router or modem is it? And test connection straight, modem/router to xbox, ? Problem is powerline
adapter and/or network configuration via dhcp... I think powerline adapter doesn't paired...

I connect the console to router by wire.(ethernet) and my PC was connected to a static IP which was assigned to my router. The stream works fine if the console and PC stay within the same LAN. However, the stream function by internet does not work since
theast update.

No Firewall or Antivirus problems. I couldn't see a wired connection on my pc in the list of devices, so I thought my pc was unaware of the ethernet connection. But, after plugging in the ethernet, confirming my xbox one was open for setup with any device
and going through the instructions for connecting to stream it worked fine.

If you plug in an Ethernet cable to your PC, the WiFi connection will stop on your PC because it should only be using the Ethernet connection. The only reason that I can think of for WiFi connecting and dropping is a problem with your Ethernet cable. If
changing the Ethernet cable doesn't work,
support.xbox.com/.../connect-live-using-pc try looking at that link to see if it can help you with your shared connection setup. Sorry I can't be of my help.