Oh, then as a serious answer haha
1) the white rabbit is just a nickname for a role-playing that they both have (the purple eyes guy and the MC) just as a theory I think he use this "game" as a metaphor since he said that the MC reminded him of his white rabbit that he lost a while ago, I think he said this because he may be the one who sleep with him that night. Since they never saw each other again thats why he said that the white rabbit was "lost".
2) we don't know yet lol.
3)probably the purple eye prince or maybe the duke on chapter 6 but I really doubt it.
4)well with the answers of 1 and 3 I guess you can guess who do I think the father is haha.
1)Who is white rabbit?
2) who is the father?
3)who fucked him that night?
4)who do you think will be the baby’s father?