I have not been able to open the GNOME shell on every distro I have tried but one. In each case, it boots to the GNOME background with no menus or other objects drawn. The one exception is Debian Jessie, but I would like something with a newer kernel. At the moment, I am trying to get Ubuntu GNOME 16.04.2 to run.

At first, I thought it was a Nvidia issue since I have a dual graphics set-up with an Intel card and a Nvidia Geforce 970. However, when I installed Nvidia proprietary driver 375.39, the problem did not go away. So I ran "ps - e | grep X" to see if the X server was running, and it is. Then I ran "wmctrl -m" to see if the window manager was running, and it gave me "Cannot open display." That led me to the discovery that neither my DISPLAY nor XAUTHORITY variables had been set. I exported both of them and ran wmctrl again. This time, it gave me the following:

Not sure if all those "N/As" were significant, I ran "gnome-shell --replace". The output was as follows:

I feel like I might finally be making some progress in figuring out why GNOME won't run on my machine, but I still don't understand a) why the DISPLAY and XAUTHORITY variables were both blank after booting, and b) why Clutter can't be initialized. Anyone know?

The presence of mutter-> clutter in the error message suggests you're required to use Wayland/Weston as your display manager in order to get gome-shell to work so I'd revert the changes I'd made and restart the display manager normally.
Then, from a terminal (konsole, for example), type in the init string. /usr/bin/gnome-terminal

Gnome terminal could start...
or perhaps fail....

However if it does fail the terminal output will give us some diagnostic information on why and we would be able to more easily troubleshoot the problem (assuming you posted the output in any follow up question)

Hope that helps.

Thanks for your suggestion! It sent me in a different direction which looks like it might be easier. When you mentioned Wayland, which I had not thought about as a possibility, I decided it might be useful to switch to Ubuntu GNOME 16.10 since Wayland support is supposed to be better with it than with 16.04. Unfortunately, the 16.10 live CD wouldn't even boot. That gave me doubts about continuing with Ubuntu. I am distro-agnostic. I just want to find something that will run GNOME on my machine. Having heard about Fedora and Wayland, I decided to try a Fedora 25 live CD, and it ran fine. So I went ahead and installed it on another partition, and after encountering some troubles with nouveau that were fixed by adding nomodeset to the grub line, I was able to update, boot, and run GNOME with no issues.

The challenge then was to install a Nvidia driver. I read up on how to do it, and put in the 378 driver. It seems to be installed OK judging from lspci output, but now it boots to the GNOME background screen and stalls before the log-in. I realized I can boot using either Wayland or Xorg by editing /etc/gdm/custom.conf, and the same issue arises with both. Noticing that it complains about acpi, I also tried adding acpi=off to the grub line, but that didn't make any difference. So that's where I am now. I have switched from Ubuntu to Fedora; I have a Nvidia driver installed; I have booted using both Wayland and Xorg; and I am stuck at a grey screen. My grub line looks like this:

Well, since you're not paying any attention to what I'm asking you to do... best of luck in all your endeavors and enjoy the view.