Learn English – What are exchanges like “How are you,” “I’m fine,” and “See you later” called

I am fine, thank you. And you? is still used, but I consider it to be overly formal. It could also be considered very polite, however, and I would probably use a phrase similar to this as a response when being introduced to someone older than me, like a friend's parent or grandparent, for instance.

In the US, the short version of this phrase is a common response, especially with people you already know or see often: Fine, thanks. You?

Either of these is fine, although if you're going to use and there should be a comma. The second sentence has a silent you in it, referring back to the fact that it was the original person who asked first and is being thanked.

I am well, thank you, and you?

I am well, thank you, and (you) yourself?

However, asking, "How are you?" may well be derived from an old greeting, "How do you do?"

According to Stephen Fry, the only correct response to "How do you do?" is, "How do you do?". Since Stephen Fry is, of course, the authoritative source of all things English, perhaps we're both wrong.