If you want to chat with her and she is happy to be 'friends', then just chat as if she were your friend.
Don't pressure her to be more than that.
Talk about hobbies, music, films, school subjects, etc
Eg:
What is your favourite ...?
Do you like ...?
Have you ever been to ...?
It may be that your close friend thought that it was a bit impolite to demand her full name on Facebook, if she hasn't put it on there herself ~ but why don't you just ask him?
Why did he feel that he should step into that conversation?
Did he feel that you were doing something wrong?
Does he like her himself?!
And, rather than using your cousin and the mutual friend as intermediaries, why not just ask her how she feels?
It could work out best ~ even if you don't get the reply that you want.
She must know that you like her, but she has indicated that the most she would be happy with is friendship.
However, you would like to know if there is any chance that she would consider going out with you.
You could just go and ask her, or you could develop a friendship with her, first, by just being pleasant and friendly ~ and chatting about what is going on locally, and likes & dislikes, etc.
Good luck
