Karen was born into the Stadtfeld family a wealthy and quite known family from what Rivalz said. I think her father managed to hide her "Eleven" descent.
There isn't much information about her we've never seen her father.
Karen admires and respects Zero for his deeds making the impossible possible
and also Zero did allow her to back out of the order if she wanted to when she was confused about her actions bringing the death of Shirley's father I think she appreciated his gentleness when he offered her an exit door
As for her feelings it's quite hard to know if she sees Lelouch in a romantic light
she did make a promise with her mother though to change the world so she might accept Lelouch's "manipulation"