Is it legal to install and use Photoshop CS6 with Photoshop Extended CS6 license?
Cannot find any answer this question.
Typically whether you get the extended or standard version is determined by the serial number. How are you getting the option to install the standard version using an extended license? If its not an option provided through the software itself then its not legal.
Photoshop CS6 Extended is extra options on top of Photoshop CS6... They're not separate programs, just additional features turned on or off (enabled or disabled) by the licensing as David said.
So not sure why you would want or need to do it anyway, but the edition is fixed by the serial number so you can't really "downgrade."