Hi Steve,
She may be doing all of these things because she misses having you as a friend.
After you asked her out in February and she told you that she only sees you as a friend, your relationship with her hasn’t been as close.
She may miss that.
She may have invited you to play cards and offered you cake because she wants the both of you to become friendly again.
When she stares at you and looks away when you try to look back at her, those may be moments where she is thinking about the moment that you asked her out.
She may be daydreaming when she does this as she recalls that moment.
This may also lead to her thinking about how much her relationship with you has deteriorated since she turned you down.
She asked the friend to ask you if you needed anything for the science event instead of asking you directly because she may feel awkward about her current relationship with you.
She doesn’t know how far she can go or whether you would even entertain her wanting to be nice and caring towards you.
In other words, she may be worried that you dislike or detest her due to the fact that she turned you down when you asked her out.
What she seems to want is to become a real friend to you again.
That may be why she is doing all these things.
She may want to go back to how it used to be between the both of you before you asked her out.