Is it possible to auto-populate a Community Builder field (using the Drop Down Single select) with the Joomla Category field?
No, we have no such fieldtype plugin.
It seems like the Conditionals extensions would be able to do this.
No, it shows/hides fields/tabs based off conditions supplied. It will not populate a field with values.
Joomla categories should be relatively static though. So simply creating a select fieldtype then manually populating it with your categories should work fine unless you intend to add new categories daily.