Is there a way for the VNC server to start automatically every time the player is started?

There's the generic pre-init system, or you could use the earlier way of replacing the binary init-file with a script that does the same thing as the binary init-file (mounts drives, starts player) as well as start the VNC server.

What is involved in getting one of those tiny computers to turn itself on/off together with the Empeg, and then automatically set itself to do the only job it needs

You need hardware that automaticaly boots when power is applied - I assume that these small computers can do that, but haven't played with them myself.
Then you need to set up the init-files so that the wanted software starts at boot - shouldn't be a problem at all - normally lots of services autostart on boot on a Linux system, disable the ones you don't want/need and add the VNC..
You'd probably want you drives mounted read-only, as you otherwise would get into problems on sudden power removal - you'd probably want to disable logging so that logging has no need to write to disk.
But nothing dictates (except good sense ) that you use Linux - if you're more comfortable setting up a Windows system to do the single task of presenting the VNC screen, then use that...

Does it need to be ethernet for VNC? I've found that kind of hard to set up.

There might (I don't know) be ways to run it over USB - serial would be rather bandwidth-limited - but I believe ethernet would be the easiest.
Use fixed IP-addresses for both (like 192.168.0.1 and 192.168.0.2) and there should be very few problems - there's no advanced routing, , DHCP servers, switches etc involved - it's just computer to computer via a crossed ethernet wire...

I've probably missed lots of things, others will surely add more info. If you run into specific problems, this BBS will no doubt be here to assist you to overcome them.

/Michael