In English I would take 'grown ups' to be a more informal phrase than 'adults' (a word a child might use for instance), is there a Swedish equivalent, or doesn't it matter?
So let me get this straight. Vuxen is an adjective, and its utrum, neutrum and plural/definite forms are vuxen, vuxet, and vuxna, respectively; and vuxen can be used as a noun, whose plural is vuxna?
If so, then what are the singular and plural definite forms of the noun?
The noun is en vuxen, den vuxne (also possible: den vuxna)
and plural vuxna, de vuxna.
Oh wow, this is very new and strange to me, especially the definite forms, although it makes sense because it's an adjective.
Just a question, does this apply to almost any adjective? How do one say 'the rich'? En rika, den rika or something like that?
I can't speak for all alternatives, but for "rik" it only applies to definite form (den rike/rika, de rika), but it doesn't work in the indefinite any more than "a rich" works in English. Just like the English equivalent, you'd replace it with "en rik man" (a rich man), "en rik kvinna" (a rich woman), or any other noun of your choice.
Why is it not vuxnarna or something for the adults? Is there a grammatical reason this word gets its own definite article, or just a random thing that developed with the language?
It's a pretty big grammatical category, 'adjectival nouns', in Swedish ett substantiverat adjektiv. We simply take an adjective and use it as a noun. You do in English too, when you say the poor in 'give to the poor' for instance. 'An adult' is en vuxen.
I used an article because I translated what Laski-Julle said, The adults, but de vuxna is just the determinate form of vuxna. The sentence on top here is about adults in general so we don't use an article, but if we wanted to say about some specific adults that they're crying, it would be The adults are crying in English and De vuxna gråter in Swedish, so it works the same way.