Major spoilers ahead. If you didn't finish the show, please don't read.
Hey! I finished Cruel Romance today and this is what I think:
1) There is no exact moment when he does. I think he is initially fascinated by her brazenness and the fact that she doesn't know who he is. This means she acts very naturally around him. So I think that we were supposed to witness a very slow progression from being slightly amused and interested, then fascinated, interested in, and finally falling for her. Seems to me like it was a mixed interest/owing her for doing him a favor type of thing at the beginning.
2) This made no sense to me. I initially thought that maybe he thought he will be doing a poor girl a favor if he bought something she owns for a high price, but I don't really know.
3) My understanding is that in those times a man and woman living together made a decision to be a husband and wife, and lived accordingly. Jinxiu and Ying Dong lived together, so they were naturally seen as a married couple. I think he just tried to make it official by asking. Mingzu and and Han Chaun didn't live together. When he asked to move in, he seemed to suggest that she would be his legal wife. When Zuo Zhen is accused of murdering the master, Jin Xiu tells Maeda that she is ZZ's wife even though there was no official ceremony at any point.
4) I didn't think about it at all, to be honest.