] "The master of the ceremonies introduced to her a very gentlemanlike young man as a partner; - his name was Tilney."

] Would Henry have asked for the introduction, or would Mr King hae taken it upon himself? And how did he do that if he didn't already know Catherine's name?

One of the duties of the master of ceremonies was to keep the event moving, for example introducing partners to encourage the dancing. I would imagine the master of ceremonies was free from the usual protocols by virtue of his role and could introduce any person present to another as he saw fit.