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You can add either. If you add example.com in the list then BOTH example.com and www.example.com are allowed. If you add www.example.com in the list then BOTH example.com and www.example.com are allowed. I know I have put that in the documentation, please read it :)

Well, yes. The documentation for allowed domains only applies for allowed domains. So it doesn't matter if you use www or non-www version of the domain, BOTH will be allowed automatically.

No. Subdomains would have to be explicitly stated. However, this would only make sense if the subdomains are served by the same Joomla installation as the one where Admin Tools is installed.

For example, if you are serving example.com, www.example.com, foo.example.com and bar.example.com from the same Joomla installation you would need to enter:

example.com, foo.example.com, bar.example.com

This is an extremely rare use case, though. The only practical use cases I have seen is using a third party plugin to have per-language subdomains or domains (e.g. fr.example.com or example.com.fr showing the same site as example.com/fr, the latter being the canonical URL generated by Joomla's language switcher system plugin); or showing a different menu as the root menu for each subdomain / domain using a third party plugin (effectively acting the same as WordPress' multisite feature; yes, these plugins have existed since Joomla 1.0, before WordPress implemented multisite!). These rare use cases are the reason you are allowed to enter more than one domain or subdomain names in this field.

If each subdomain is served by its own Joomla installation you MUST NOT add the subdomains in the main site's Admin Tools configuration. You should only add each subdomain to the Admin Tools configuration installed in the respective subdomain's Joomla installation.