Well, in a way you are actually right. "Äpplena" is definite and the "de" also makes it definite as well.
This phenomenon is characteristic to the Swedish language (and mostly also to Norwegian bokmål, I think) and is a distinction from e.g. Danish. You might find some information on the topic by looking up 'double determination Swedish' on google.
I think this will appear in the lessons on adjectives but here is how it generally works:
"a dog" - "en hund"
"a white dog" - "en vit hund"
"the dog" - "hunden"
"the white dog" - "den vita hunden"
He had to put "den" because "vita" is an attributive adjective (basically an adjective closely attached to the noun it refers to, without a verb) and that requires you to use a proper definite article.
There must be an agreement between the definite article we use and the attributive adjective it refers to. Keep in mind that, sometimes, the definite form of such an adjective might be the same for the plural (which is the case for "vita"). I found this confusing the first times but, of course, it all comes down to practice and habit.
WRAP UP: since "the dog" is singular and "white" is an attributive adjective you have to use a definite article that agrees with "white". The definite form of "vit" is "vita" and the definite article is "den", so the sentence is "den vita hunden"
I think this is more just that two different words happen to be the same. It happens in all languages I can think of at some point, but yeah, it's confusing.
So de can either be a form of the article, as in den, det, de, or a pronoun denoting several people, as in vi, ni, de.
At least we don't have the same word for both singular and plural you and their object forms, as they do in English.
There are two words "de" – one is the pronoun meaning 'they' and the other one is the plural of the article which is det or den in the singular.
It doesn't introduce a subject here, it's just there because we need an article before adjective + definite noun, just like we would have said det gula äpplet 'the yellow apple' or den gula boken 'the yellow book'.
No. The article is only left out in constructions that are more or less names, or fixed expressions. Gula änkan 'The yellow widow' is the name of the champagne brand Veuve Clicquot and it's a name, so the article is left out. As far as I know, there's nothing called Gula äpplena, so you cannot leave out the article.
If I'm not mistaken, this is what I take from all the helpful comments:
ett äpple (singular) ett gult äpple (singular) Det gula äpplet (definite singular) De gula äpplena (definite plural)
What I am missing is, is (indefinite) plural: De gula äpplen <- Is this correct? When I google it, I don't see that as an option at all.