downloaded High Sierra 10.13.6 update, clicked Restart NOW, but the restart never completes. I've waited overnight, with the Macbook Air plugged in, and the only solution is to hold down the power button to force a shutdown. HELP

Try restarting in Safe Mode and then installing.

Use safe mode to isolate issues with your Mac - Apple Support

something like that happened to me, but I think the install stopped halfway when I restarted my MacBook Air. Since then it often won't wake from sleep. I have started up in safe mode, which didn't help.

Not always, but I can tell the computer's on but the login screen won't come up no matter which key or key combo I try. Eventually I have to manually shut down and restart. Sometimes that works and sometimes I have to do it twice.

WHat is going on?

also, it appears that the update has installed fine. "About this Mac" says the OS is 10.13.6

But running disk utility after the attempt to wake up and restart, I get the following error message: