I tell my 37 year old coworker that one of our student staff guessed that she was 44 years old
Why would you do that? Have you stopped to think about how her being perceived as being older would make her feel?
She likes to lie about the fact that she's 37 years old, but I don't give a shit about her
You should be considerate of your co-workers feelings. Telling them things just to upset them is cruel. Making fun of someone just creates tension in the work place for everyone.
She made a big deal about being 37, so it's payback time.
You'll be working with this coworker for a very long time and not caring about their feelings will make this work very awkward. While she may have annoyed you, do you think acting this way will make anything better?