I had to edit your question as it asked more than one question. Please remember in future that enotes does not permit multiple questions to be asked.

You are right in identifying that as Miss Brill sits and enjoys the Sunday afternoon she suddenly believes that she is part of a play. However, we are never told specifically what role she has, apart from being there every week and sitting on the same bench and watching the same people. Note what she says:

They weren't only the audience, not only looking on; they were acting. Even she had a part and came every Sunday. No doubt somebody would have noticed if she hadn't been there; she was part of the performance, after all.

We can see that this dream of hers is designed to give her significance and importance in a life that, as we discover, gives neither to her. Particularly significant is her feeling that it would be "noticed" if she didn't turn up. As we see at the end, she lives a meaningless life in "her room like a cupboard" and, despite her belief that she was an actor and not just part of the audience, it is clear that for Miss Brill life is nothing more than a spectator sport.