She fantasizes about a prince and starts receiving letters in the following chapters, so I think it's possible she let the duke know she wants to find love and get married. So marriage prospects and invites are flying in. He's quite shocked that his dour daughter is suddenly so 'childish'. But it can be seen that he orders the butler to look after her etc. in one frame after he leaves that room. I hope this helps :)
It does seem like he has nightmares and it's of his daughter being beheaded. He probably treats them as merely a dream but it's too real for him to ignore it so he notes the changes happening in this timeline. Eventually he may realize his dream was in fact his previous life. The other thing, Keira is found guilty in her past and he hated her enough to sentence her to death, in this life however, he seems to care for her a bit more. That leads me to believe he knows more about what happened after she died and he feels guilty for wrongly accusing her. Though its the dream version he wronged, he can't seem to separate dream from reality so while he doesn't want to interact with her like in the past, he can't seem to stop himself from asking about her wellbeing....anyway, thats the general gist of what im getting.
Can someone tell me what happened at chapter 8-9 where The Grand duke looked like he's depressed/stressed. What did the MC tell him that made him shock?