Does USB and Ethernet come off the same chip?
No.
in terms of transmitting audio from USB into the processor
This is not a normal thing to do. The USB connection is only for serial data.
package it up and send it out the ethernet port
Again this is not normal.
The Ether net board will support a UDP ethernet package both in and out. It is up to you what you put in these packages.
I think this stuff all works at a much lower level than you are thinking.