Affably Evil: Avo could qualify as this, especially Storm Surge. He's actually a very calm, intelligent, and entertaining individual... who nevertheless destroys people's buildings and structures. It helps that he's actually become friends with some of the people he's griefed (I.E. the people of Spacemonkey and Doridian).
Always a Bigger Fish: The true damage to the servers don't come from Avo themselves, but rather from all of their fans that see the videos and come onto the servers after watching the videos.
This may be part of why Team Avo doesn't usually put the IP of the servers they grief in their videos.
Anti-Villain: Although the entire point of their videos is to burn stuff, kill people, and aggravate the other players, they do target admins who have an overinflated ego or have otherwise made themselves targets, as with AllNaturalX. Not only that, they're now friends with two targets of their griefing, Spacemonkey and Doridian. In addition, they will also leave you alone if you say 'Team Avo deserves Ice Cream', which apparently translates to 'I/we appreciate the effort you put into this'. Inversely, if you say they don't deserve ice cream, you are screwed.
Arch-Enemy: The fully-Christian server they trolled have gone on to declare them the worst thing ever to have happened, and gone to very extreme countermeasures to make access for them impossible. Apparently, this is in part owing to them opening the floodgates for other trolls.
Nice Job Breaking It, Hero: After stringing along the above Dirty Old Man, they have him believing they are a hot girl sending him pics - when one of the members' invisibility runs out. The others quickly turn him invisible again, but it's too late. The victim sees his name - "Agent Harry" - and realizes they've been messing with him.
Running Gag: The aforementioned destruction of every single boat they find.
The Smart Guy: Krysk, Team Avolition's main programmer for their client.
Stuff Blowing Up: One of the ways they grief when they find a server that allows TNT and explosion damage.
There Is No Kill Like Overkill: Any server with Big Brother requires that they do this; Big Brother records who modified which block last, so doing something like putting lava in a house would just get them banned. Their solution, therefore, is to utterly destroy any structure and remove any trace of its existence.
First, they demolished a castle. As they were trying to bury the remains, they received help from the player who built it.
Second, Storm_Surge was destroying a gazebo. He was caught in the act by another player, so he pretended that he was "upgrading the floor." He asked the player if he could fetch some logs. After the new structure was built, the player helped Storm_Surge protect it.