(1) can work with the singular or it only works with the singular?
It all depends on how we understand the phrases, which in in turn depends on the context.
1. He is the only one of those men who is/are always courteous.
It is true that the singular, contrary to what I said in my first reply, will be more usual.
However, suppose you are speaking with a friend or a colleague about what some people are like:
A: There are men who are always courteous.
B: And what do you think of John?
A: Well, of all the people here I have to say he is the only one of those men who are always courteous. (meaning: he is the only one of those men you mentioned as being always courteous, that is, he is the only one who belongs to the courteous kind.)
Using the singular or the plural will depend on where we can "break" the phrase:
1a. He is the only one // of those men who are always courteous.
1b. He is the only one of those men // who is always courteous. (Among those men, he is the only one who is courteous.)
In the other examples, there was -- I think -- only one possible "break":
2. He is one // of those men who are always courteous.
3. Pope Francis is one // of the popes who have led the Catholic Church for almost two thousand years.
4. John is the only one of my friends // who likes classical music.
5. John is one // of those friends of mine who like classical music.