I am only allowed to respond to one question, so you will have to access the enotes study section on this tremendous short story to find out the answer to your second question. Also, you might want to check the Question and Answer section of this group as I am sure you will find an answer there to your question about the character of Miss Brill.
This short story then presents us with the thoughts of Miss Brill as she goes to the park on a Sunday. The story is narrated to us using the stream of consciousness point of view, which allows us to "hear" each thought as it comes into Miss Brill's mind. She talks about the pride she feels in wearing her "fur" again after a while and how much she loves it. She then comments on the various people in the park and the music that she hears. It is clear that she comes every week at the same time and enjoys watching people and listening in on their conversations. She loves "sitting in other people's lives just for a minute while they talked round her." Some of the people that she watches who also come every week sit "like statues" in silence, which makes Miss Brill imagine that they have come out form "dark rooms or even--even cupboards!"
As she carries on enjoying the day and observing she imagines that she is watching a drama that involves everyone, even herself. This gives her an immense feeling of importance and significance. However, a young couple come and sit close by to her bench and make fun of her, referring to her fur as "fried whiting" and calling her an "old mug." Miss Brill goes back home, not buying her normal weekly treat of honey cake, returning to her "little dark room--her room like a cupboard" and quickly putting her fur that she was so proud of away in its box. As the story closes she seems to hear something crying.