I agree with your first theory cause her face wasn't shown so it's clearly alluding that she's the mother it also might be that her and the duke did like each other but because of the count they might have forgotten each other and their child's existence thus leaving the count to do whatever he wanted with her also he probably erased her memories to cause she can't remember past 7 years old which is the time she probably spent with both parents and as you can see in her flashback the count was asking if she remembers anything meaning he probably already knew also there's alot of evidence that someone tempered with their memories and that the count being jealous probably took part in it also the fact that no one knew means that the duke knowing the greed people have against his family probably hid the fact that they were dating also FL was told by the count in one of her flashbacks that the duke was a woman snatching trash alluding that he was indirectly telling her that her mother and the Duke knew each other and that the duke probably had her attention before him and he wanted to adopt her who looked alot like ther duke just to torture her for his own selfishness
I also think there are a lot of signs leading to the count being responsible for them forgetting things, that is why I think my first theory might be the most possible answer, but it could be anything as long as they are father and daughter cause I really don't like the idea of being siblings or any other form of being relatives.
This story is cute and all but its very confusing and I can't wait for everything to be explained, and I think most of the other readers can't wait too, so I'll be sharing my theories as to why they are related.
1. Her mother is that woman the Count liked but did not like him back. The reason she did not like the count is because she and the duke like each other, that is also why they had a child which is Hailyn. The count knew about this so he made them forget their child and sent her to an orphanage so he can adopt her, the reason I say "them" is because I'm not sure whether the woman the count liked is dead or alive.
2. Hailyn might be her sister, cause it wasn't clearly shown what the duke's father did the last eight years even though they were "supposed" to investigate it.
3. Someone strong from the Holy City took some of the duke's power and used it to make a random woman pregnant with his child, so they could use the child with Kalisto blood because of her holy power, but like some mishaps happened and she ends up at the orphanage.
That's all, just wanted to share this.