Not in my part of the United States, New England. :) You should ask either "How's it going?" or "How are you doing?" instead. If you are close friends with the person, you can even put a vocative expression at the end; for instance, "How's it going, bud?", "How's it going, buddy?", "How's it going, dude?", etc.

Yes, occasionally. It's quite polite and old-fashioned, but still used sometimes in formal contexts. But there are a few important things to understand about 'How do you do?': 1. It isn't an enquiry. 2. The response is not 'Fine' or 'Very well' or anything else of that nature. In fact, when someone says 'How do you do?', the only way to answer is to repeat 'How do you do?'. It's the same as when you say 'Hi' or 'Goodbye'. 3. It is NOT an alternative to 'How are you?' or 'How are you doing?' or any other everyday greeting. In fact, the only time it is appropriate to use 'How do you do?' is when you meet, or are introduced to, someone for the first time. For example. A: Mr Brown, this is Mr Smith. B: How do you do, Mr Smith? C: How do you do? As you can see, the function of this exchange is the same as 'Hello' / 'Hello' or 'I'm pleased to meet you.' /'I'm pleased to meet you,too.'

You can still say this. It just sounds more polite and proper and is less commonly used but there is nothing wrong with saying this.

Sometimes, but it is a bit formal sounding and not too common anymore. More common, and more casual is to say "how are you doing?" or "how is it going?"