Regedit wont run, and grant admin full control context menu wont work

This looks like a file permissions problem. You should be logged in as Administrator to correct this.

Open C:\Windows, locate regedit.exe file and right click on the file and select properties.

Go to Security tab and check the administrator's rights for this program. If it is not full control, click on Advanced, go to Owner tab and change owner to Administrator. Close the dialog box and change the Administrator's rights to "Full Control". Now hopefully you should be able to change the compatibility mode from Win 95 to Win 7 and execute the program.

Permissions - Allow or Deny Users and Groups

Or alternatively you may search on Google and download one of the free registry editors like

Registrar Registry Manager 6.52
or
O&O RegEditor
or
any other freely downloadable registry editors for window 7 and use.

What happens if you boot to Safe Mode and login as Administrator.

Note that you have to enable the hidden Administrator account first.

Open a Command Prompt
To enable: Enter net user administrator /active:yes
To disable: Enter net user administrator /active:no

Your Compatibility tab is ok, that's the way it should be. Show me 3 pictures of your Security tab ( Administrators and Users highlighted, and also the Advanced tab).

In regedit Properties, click on Help me choose the settings, see if that gets you any where.

Thanks for pictures. Looks ok, you have the permissions ok. Have you tried booting into Safe Mode and trying to run regedit?

If you can't run regedit in Safe Mode, it's quite possible your regedit.exe is corrupt.

You might try doing a SFC:

System File Checker

SFC /SCANNOW Command - System File Checker

SFC /SCANNOW : Run in Command Prompt at Boot