1) Give users an mail-adress, i.e userid@mydomain.com upon registration to the community I administrate

Joomla requires email address as it is one of the core fields; thus CB too requires it and asks for it at registration. The field responsible is email found in Field Management. If creating users in backend you'll need to provide an email as well so yes you can specify emails for users.

2) Make this a fully functioning mail-adress that is ONLY accessible when logged in to the forum?

No, CB does have a setting whether to allow emailing of users or not, but is not conditional as such. Please see General tab of Joomla Backend > Components > Configuration. Additional you could use document subscriber CB Privacy to allow users to control the privacy setting for their Email field.

You would likely need a complete and complex new CB plugin to accomplish this unless you intended to create each user manually as your mailserver will not simply create email addresses automatically without secure code doing it. I recommend letting your users provide their emails rather then creating ones for them. At any rate, yes will need to be done by hand or via very extensive and complex code (likely need a contract developer and contact with your mailserver host).