I like all of the perspectives that have been brought out so far, and just want to add another. Henry engaged Catherine to dance with the expectation that he would then have her undivided attention. When she is drawn into conversation with JT, Henry is feeling left out. So in addition to being flirty, and maybe trying to probe Catherine's ideas on the subject, I've always read the marriage/dancing analogy as being his way of conveying his expectations for their time together in the dance, and his hopes that she will give him her full attention and appreciate his, as he expected when he asked her to dance.