First for the backup, all mailboxes are part of the backup and thus can be restored. They are part of the platform backup or in Cloudron terms "box" backups.
For the role, there is currently no specific email role and the user manager role also does not apply here. So far we haven't seen much requests regarding those roles, so there is a lot to be improved I think. Also while reading your question, it may actually make a lot of sense to allow the usermanager role to also manage mail related things. Not sure yet, feedback welcome.
To answer your immediate question, the only way to do this is to grant admin rights for this user at the moment.