Sounds like you have your mind made up but to answer your questions:
An official can absolutely address the crowd and have a fan removed if needed. Some officials are better at it than others and I would prefer they get game administration involved but that is not always an option at that very moment.
Not necessarily on the injured player, if the coach comes onto the court to attend to a injured player then the player will have to come out until the next dead ball. If the player says, I am okay and the coach does not come onto the court, the player can continue to play.