If the Skinner character doesn't have An Arresting Skinner, they don't have the right tool to stop an angry mob dead in its tracks, I think, but jumping into a tense situation (that isn't quite an angry mob yet) and defusing it with a performance seems like classic Skinner behavior to me. If the player rolls well, let the move play out. Maybe the crowd shuts up and listens, and now it's waiting for the Skinner to say something to the crowd instead. Afterwards, you can have the NPC hardholder chew out the Brainer and say that he/she would have been killed if the Skinner hadn't stepped in... always be building triangles and splitting the players apart.
The answer to your second question is no, it really goes counter to the rules and spirit of Apocalypse World to try to structure situations to force a certain outcome. If there's an angry crowd, let whichever PC is there use their Moves and connections to deal with it as best they can -- for the Skinner it might be a performance, the Brainer might start brain-puppeting the ringleaders, a Gunlugger can shoot over the heads of the crowd, and an Angel might just have to say that's he/she isn't going to tend the wounds of anybody so stupid as to start a riot at the hardholder's palace and maybe that would be enough. Just let it play out and take those events and throw them back at the players, hard.