Yes, only for a first meeting, and a very formal one at that. If someone said "how do you do?" to me, I'd say "good to meet you" in response.

The correct usage is to use it the first time you meet someone in a formal situation. Only in that context so no, it's not synonymous with 'how are you?' If someone says it to you, you repeat it back to them like so: How do you do Zahra? How do you do Paul? That being said, I never use it. The situations in which you might use it are not exactly very common, but even then I would probably say something like 'pleased to meet you' or 'nice to meet you' as you wrote in your question instead.

It's an extremely formal greeting, 99.9% of English people would never use it, unless you are part of the royal family or high society you would never use it! It's definately not interchangeable with the others, way too formal!

HOW DO YOU DO? , is an old-fashioned and more formal way of saying HOW ARE YOU? It is seldomly used in spoken English today.