1. More or less, that's what it is doing. When the Plex server is running, "sleep" just turns off the HDMI output from the ShieldTV and possibly suspends other things only needed when it's active.
2. You can try to turn off "CEC" so the Shield doesn't get communication that the TV has switched away.
3. CEC sometimes goes by different branded names. Samsung calls it AnyNet+, Sony calls it BRAVIA Sync, etc. (you can find a list of branded names here ). HDMI-CEC allows your devices to communicate to each other through the HDMI interface. For instance, pressing a button on your ShieldTV remote may turn the TV on and set it to that input, where switching inputs on your TV may make the ShieldTV sleep. They can also communicate volume adjustments.
4. The IR only works for volume adjustments. You would need HDMI-CEC enabled so that the ShieldTV can turn the TV on when you press a button on the ShieldTV remote.

1. Sleeping the device will allow the Plex Server to continue running in the background.
2. Not sure, we use our shield for everything
3. CEC (Consumer Electronics Control) allows devices to communicate to each other through HDMI. Depending on the devices used, the Shield can tell your TV and Receiver or Soundbar to turn on or off.
4. If you've setup IR instead of HDMI-CEC, just point the remote at the TV and press the home button. I believe any others may work.

Dumb question, but can you turn the Shield on and off(or even just sleep) via the included remote? With CEC on, that should also turn the TV on and off. I'm just wondering, as that's the only thing holding me up from buying a Shield, I read that the remote has no power button so you had to hit the power button on the Shield itself to turn it on?

If I'm running a plex server I obviously want it running even when nobody is watching on the nvidia box. Is this just turning off "put device to sleep"?

You don't even need to do that. As long as the Shield is powered on, the Plex server will be running and responsive, even if the Shield is in "sleep" mode. Keep in mind, sleep mode on the Shield isn't quite the same as sleep mode on a PC.

I notice if I change from the HDMI input for the shield to, say, cable, it stops playing. Is this avoidable (in my case, watching sports on both and wanted switch between them)?

This, I don't really have a good answer to. I haven't done much with that functionality.

What exactly is CEC TV on/off?

I set up the IR for my TV. How do I actually now use the nvidia remote to turn TV on/off?

These questions actually answer each other. You can set up the Shield to use CEC to turn your TV off and on (as long as your TV supports CEC). I don't recall the exact options off the top of my head, but my Shield is currently configured to automatically turn on the TV and switch the active input over when it comes out of sleep mode, automatically come out of sleep mode when the input becomes active, and turn off the TV when it goes into sleep mode. If you have these options enabled, and have CEC enabled on your TV (this will vary by your TV's manufacturer and model), you can turn off your TV by going to the home screen, long pressing the Back button, and selecting Sleep Now from the menu that comes up.