"Hi!" is the short form of "Hello!". Hi! is more unformal than "Hello!" and as a formal greetings you also have "Good morning!", "Good afternoon!" and "Good evening!". "How do you do?" is also a more formal question. English people and americans usually say Hello! / Hi! before asking you how are you.

Yes, it is. Very formal and not used by at least ninety percent of the UK population. Forty or fifty years ago it was very popular and I imagine that most of the population used it. It was used when you met someone for the first time. When I was a kid, you had to use this expression with everyone you met.

Very formal and used in business introductions, "this is John, you'll be working with him", "how do you do" as you shake hands.

A super posh person may also say "Pleased to make your acquaintance" but not really in real life! Nice to know though!

Yes it is also the same as !How are you!