der Apfel = singular, die Äpfel = plural. In the dative case, like the example, the plural would be "den Äpfel"
Not quite: “einem” is dative for masculine and neuter gender (einem Mann, einem Kind) and einer is dative for feminine gender (einer Frau). In plural, you don’t have any form of “ein” since the indefinite article is used only in singular: It is the same as in English – “ein” (a) carries also a meaning of “one”. If you want to use a plural indefinite noun, just put it down without an article. But don’t forget that German nouns take -n in dative plural (Kindern).
Isn't it an umlaut thing? "Apple" singular has no umlaut,. whilst the plural does. I tend to umlaut plurals (it's a newbie thing) as, some 70% of the zeit I'll be richtig (What's German for "ish"?)
that's partly it, but also the Der / Die thing is a clue. In this case, "dem" is the dative version of der... so it's der, not die, meaning it's singular.
So mit is a dative preposition and all words that follow it must be dative pronouns?
The objects that follow it (not necessarily pronouns) have to be in the dative case, yes.