Run cmd.exe in Administrator mode without explorer.exe
I found a simple solution that I can't believe I overlooked. Just open Task Manager, click "Show processes from all users", then click Continue to get Task Manager with Administrative Privileges. From there, go to File > New Task > cmd > OK, which will now open with admin privileges. From there I was able to get sfc to work just fine.
Is there any other way to run cmd.exe with admin privileges?
Yes, restart your computer in safe mode and log in as the "administrator" user, he normally only shows up when you are logging in to safe mode.
If not, is there any way to run sfc.exe without admin privileges?
SFC requires administrator privileges, however you can remove the drive and run sfc.exe on another computer that you can run software as administrator, you will need to use the arguments /OFFBOOTDIR and /OFFWINDIR to point at the drive letter of the attached drive.
sfc /scannow /OFFBOOTDIR=d:\ /OFFWINDIR=d:\windows
That is what you would type on the other computer if the new attached drive showed up as the D: drive.
If not, does anyone have solution to my original problem of explorer.exe stopping and restarting ad nauseam?
Your last option is use the System Recover Options menu and try rolling back to a restore point. (it's not SFC but it may fix the problem)