I think Henry is very careful about the people he chooses to 'hang out' with. Being the clever person he is and worldly besides, he is no doubt very good at determining those who are shallow and lacking in morals and trustworthiness pretty easily from those who are genuine and honest. He probably determines rather quickly the people whom he considers worth knowing---whether he would like to engage in future conversations with them, if he would like to dance with a young lady at another time at another ball/dance, etc.

Henry and Catherine mutually enjoy their first conversation and I believe Henry is assessing her from the start, first as to whether she would be someone he’d like to hang out with and eventually as someone he would consider marrying. His quizzes are in a way tests to see if she will make a compatible partner: is she principled? Is she honest? Can she think for herself? Is she open to learning? Is she likable? Is she fun to be with? Is she reasonably intelligent?

It’s interesting that he ‘teaches’ her by questions more so, or at least as much as, by just coming out and telling her directly what he wants her to see. He helps her to discern what is important by allowing her to realize the answers for herself. Not only is she learning lessons but he is modeling a process for reasoning, making deductions and gaining insights entirely on her own.

I believe, absolutely, that Henry is using these quizzes as time goes by to determine if Catherine can meet his expectations of what he is looking for in a wife.

I also agree that he finds her novel in the same way Darcy found Elizabeth novel. Darcy was attracted by a pair of fine eyes and Lizzie’s lively wit. For Henry it is a pair of sparkling eyes and Catherine’s refreshing honesty and genuineness. Neither man was looking to be so unexpectedly charmed.