I think it'd be a bit better to use "...annoyed by performer's annoying words." because 'annoyed at' is usually said when a requirement of some kind bothers you. E.g. "She was annoyed at having to do everything again."

In American English, we say "The audience IS annoyed at the performer's annoying words." Note that "audience" is singular here in the States, although, I have read online that "audience" is considered plural in other dialects of English. Also, we would say "The audienceS ARE annoyed at the performer's annoying words." if we were speaking about multiple audiences. For instance, the audiences at different venues throughout the United States.

Yes, it's fine. It might be a little redundant to use variations of the word "annoy" twice but it's perfectly acceptable. :)