"The new boy"
Is "chlopiec" outdated in modern Polish, even for a format dialogue? My parents always used "chlopak" instead. The former sounds very out of place to my ears.
No, it's not outdated. "Chłopiec" is a diminutive form of "chłopak" and more often is used to name small boys (kids).
It is not diminutive, those are different words. And I would say in meaning it is more like chłopak is augmentative form of chłopiec.
Chłopiec means "boy", but it is less often used for young people, who are not considered "children" anymore. Chłopak means older boy, young man , youth ; also boyfriend
I think that are almost the same level of formality, but in the formal situation, I think chłopiec still fits, while "młody mężczyzna"( young man); nastolatek (male teenager) fits better.
Also in plural "chłopak" is a tricky word, it has a declension that is complicated for Polish people. Plural nominative is "chłopaki" which needs not male personal adjectives and verb forms.
I don't consider either of them as diminutive of the other. Rather as two forms of the same word, perhaps putting „chłopiec” a bit higher on the formality scale.