I don’t even really hear “how do you do” at introductions - “nice to meet you” is the standard.
To address traditional questions of etiquette, it is advisable to consult a traditional etiquette guide. I happen to have one from 1942 right here.
- There are two rules of etiquette at play here. The first is when someone introduces you to someone else. The correct procedure is for the person to whom the introduction is being made to respond with “How do you do, Mr./Ms. Wakinyan.” Proper procedure dictates that the person who introduces you to then continue the conversation with an “ice-breaker,” like “Wakinyan is a Sweedish journalist,” which opens the door for further conversation. Note that in this case, you do not reply to “How do you do” at all.
The second instance is the self-introduction. The proper form for this is, "Excuse me, Mr. Bob? I’m Wakinyan, the Sweedish journalist. Then follow up with something that connects you to the other person: “We met in Oslo last summer,” “I spoke to you for a story I was working on,” etc.
- A formal business letter should preferably be addressed to someone. If you don’t know the name, the proper salutation is “Dear Sir,” “Dear Madam,” or Dear Sir or Madam."
Of course, this is from 1942. I also have a 1976 secretarial handbook that states that the word “madam” is passing out of style and suggests “Ladies and Gentlemen” (with no “Dear”)as a proper salutation.
In response to “how do you do?” my preferred response is, “sober, but it’s still early.”