How to configure ConEmu to run cmd.exe as an Administrator from an Explorer integration

To answer the first part of your question, when you hit the Windows key and type "CMD" you can hit Ctrl + Shift + Enter to open as administrator.

To answer the second part of your question, paste the following into notepad and save it with a ".reg" extension:

Windows Registry Editor Version 5.00

[HKEY_CLASSES_ROOT\Directory\shell\runas]
@="Open Command Window Here as Administrator"

[HKEY_CLASSES_ROOT\Directory\shell\runas\command]
@="cmd.exe /s /k pushd \"%V\""

Then run the file. It will merge the changes into the registry and add the option to your context menus. (No shift key needed.)

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.