Is the first "this" necessary in this sentence?
Yes, "this" is necessary in this case because it is referring to the specific dance and he is asking whether or not it was in the audition that he performed it. It would be awkward and incorrect to say "was in the audition", and so you need to put "this" in this sentence since he is asking a question. You could also replace "this" with "it" to refer to the dance and say, "was it in the audition you did this dance?" instead.