Hello,
like I have tried to explain at the beginning of the thread, the MooTools libraries are added by the code:
JHtml::_('behavior.framework', true);
in the index.php.
Most templates use generic calls to the Joomla framework in order to include the right libraries into them.
But that code may change depending of your template's own framework.

I would suggest you contact the template's developer as there may be something specific to the template I do not know about if you cannot use the standard way of adding the libraries.

Hello,
It is already included in the standard Joomla! templates.
Look into any beez5 or beez3 templates from Joomla! (templates/beez3/index.php for instance) and you will find that line there.
Then, if you don't see MooTools in those templates, you have installed an extension that removes them.