Not necessarily. Blackhat folks are generally divided into 2 basic groups...the type you described, and the kind that for example break into servers just to show off their skills but they don't do anything other than prove to their peers that they were in...and the more difficult the server they get into , the more praise they get from the "scene."
There are whitehat folks who do the same...the difference being blackhat don't tell the company whose server they broke into how they did it.
No, that isn't accurate. Most blackhats I've known don't steal anything or benefit at all from their conquests. They think hacking is fun, that's it.
Doing it for bragging rights is not accurate either. Plenty of them brag, sure, but they hack because they like to hack. People always want to make that more than it is.
Blackhat is outright stealing/espionage/manipulation of other devices for your own gain.