SPOILER
Actually The Empress want to test Shuli. She hated her cause she resemble the late Empress but at the same time she is curious why Johannes have faith in her.
Shulli tried to prove that she has no marital relationship with Johannes using that letter but it's not enough so she use the last resort by using Doctor and it proved that she is a virgin, but when the court about to annul the marriage, The Empress convinced by Shuli effort, going so far to tarnish her own reputation to save Jeremy, so she cancelled the trial and Shuli remain as their mother.
No. he never harbor any romantic feeling toward Shuli, they relationship is like brother and sister. In the first timeline, at first they misunderstood and hated her but after they grow up, they realized that Shuli was a good person, but their relationship already too strained so it's not easy for them to suddenly became family, it was his fiancee that made up the lie that Jeremy didn't want Shuli to attend the wedding and kick her out of the house, when he found out Jeremy almost strangled his fiance in the altar and cancelled the wedding, but it's too late, Shuli already dead.

So will she leave the family afterward or stay with them?