I'm interested in this too, I wrote my own PAL driver for UNO/Nano and I had trouble implementing other libraries in there, in the end I hacked my own things to work, I think I also didnt use any of the sound functions to avoid conflicts. There could also an interrupt issue as I think the Ethernet uses a timer somewhere
It is possible to do an awful lot of things simultaneously with video, I wrote some code to drive a VGA monitor (over twice as fast as PAL) in 17-bit colour while reading serial data at 250kbps from the UART, so I am sure a bit of Ethernet SPI should be possible
Bob