She has not spoken to her parents in about 9 years, because they were not accepting of her being trans. Her father wants to reach out and apologize for his initial reaction, but he's afraid to do so (Eloise and her father are so very important to me. He loves her very much, and he's ashamed of his reaction to something he didn't understand. All he wants is for her to be happy).
Her relationship with Agnes lasted about two years, and ended when she died. She spent quite a bit of time in mourning, became a volunteer firefighter for a few years, and kept stumbling across signs of the lightless flame. She went hunting for information about how to stop them, and this eventually lead her to Gertrude, and the institute. She arrived a few days after Gertrude's death, and ended up taking a job at the institute anyway.
She's covered in burn scars from Agnes, mostly on her face and hands, which she sports proudly (Tim asks her where the scars are from eventually and she just says 'lesbianism' and refuses to elaborate)
I wrote a statement about Agnes from Eloise's perspective that gives more information about their relationship dynamic if anyone is interested.
Also please ask any questions you'd like, I'm happy to answer!