I don't think there is a "black and white" rule there. With that out of the way, here's my opinion.
"To whom it may concern" is used where the letter is issued for use by someone without regards to who receives it, whether a person, or an institution etc.
"Dear Sir or Madam" is a more specific address.
Dear Sir or Madam (some write it Dear Sir/Madam) would be an appropriate salutation when you are writing to an institution and you don't have a name. It is in common use, at least in the UK and the EU, and is considered polite and professional.
"Sir or Madam is a respectful way to address the person when you don't know their gender - that is, when you don't know whether the person who will read the e-mail is male or female. Though it is still acceptable to write Dear Sirs in the UK, this salutation is no longer considered politically correct in the US, as it excludes women.
Dear Sir or Madam is the salutation I teach my students and business clients here in Germany. It has a courteous, respectful and professional tone to it, which is suitable for formal and more relaxed written communication. This tone of politeness and respect is very important when communicating with people of most cultures in the world.
Some people are of the opinion that you should make an effort to find out the name of the person being addressed by calling or e-mailing. If that's not possible, it is recommended to make up a title that fits the situation, as you have done; for example, Dear Student Insurance Officer.
There are many books and websites advocating one style or the other, but there is no definitive style that all agree upon in the English-speaking culture.
You seem to have thought this out well and have provided some good examples.
In the end, I can tell you that as a non-native speaker of English all of your efforts to write and express yourself in good English will be respected by the person reading your email.
Best Answer
In English, "Dear Sir or Madam" is the traditional and customary order. It does sound quaint (and sexist to some) — but's that's how it is.
You would indeed be thought "outlandish" (a good choice of term there) if you were to reverse that order.