***** NOT A TEACHER *****
This time, I am reasonably confident that the following comments (not "answers") are accurate:
1. Yes, everything is in its right place.
2. Yes, that sentence is grammatical.
a. If dear Miss Nightingale wrote it, it must be grammatical! (Just joking)
3. This kind of sentence that starts with "it" is very common.
4. In theory, your sentence could be written as:
"That it should do the sick no harm may seem a strange principle to enunciate as the first requirement in a hospital."
a. Native speakers usually do not start a sentence with a noun clause.
b. As you know, an English sentence usually starts with the subject.
c. So native speakers use a so-called dummy subject in sentences such as yours. The word "it" is the formal subject, but in reality it is a substitute for the real subject ("That it should do the sick no harm") that comes at the end of the sentence.
i. That you are intelligent is obvious.
ii. It is obvious that you are intelligent. (This way is preferred by native speakers.)
Student or Learner