I have always been interested in gender issues and feminism and one area I have studied is gender and language. As a male feminist, something I notice is the way people think the word "woman" sounds rude in some contexts, but "man" doesn't.
Because feminism is about equality between women and men, I often cringe when I hear "lady" in situations when "gentleman" isn't used. I'm not saying "gentleman" is never used, but when it is, it is only in certain contexts and often in its true sense ie. "he's a real gentleman". I have never liked this kind of unequal use.
I have heard "lady" and "gentleman" both used in polite contexts and I don't dislike that, but I have often also heard "lady" being used in ordinary conversation, when men are just "man", or "bloke"/chap" and increasingly "guy", which is perhaps the most commonly used Americanism that has come into British English.
I think the way "lady" has replaced "woman" in many situations is a good example of semantic derogation. My least favourite uses are where "lady" is used loosely as a substitute for woman, it can sound way too polite and twee in these contexts, as well as a little odd (you wouldn't call a man a gentleman), and I hate when a girl or young woman is condescendingly addressed as "young lady", I mean a boy or young man would just be called "young man" and not "young gentleman". What is is that people find rude about "woman"? Going back 100 years or more, people actually said "young woman", if you read texts from around that time.