I have reset my Router to no benefit. What is “deep sleep and wake up”?
i do agree it is probably a network problem. I have considered hiring a computer tech to sort it out for me.
Deep sleep is pushing down the power button for three seconds. Then when you push the power button again, the Core wakes up into active mode. It's effectively the same as a power off/power on and can resolve discovery issues across the network.
Regarding your setup, the Core can be wired into your network from any location through ethernet. The switch is simply taking one ethernet connection in, and sending it to a number of ethernet outs to multiple devices. You would, for example run a lead from a single ethernet port into the switch and then connect computers, Core, Apple TV's etc from individual leads connected to the ethernet out ports on the switch (the same as splitting electrical outputs on adapter boards).