Learn Hardcore German: Die Kinder spielen zusammen auf dem Spielplatz. - The children play together on the playground.

The sentence talks about more than one child, so it uses the plural form Kinder. In German, the definite article for all plural nouns in the nominative case is die, regardless of the noun’s gender. Thus die Kinder means “the children.”

Why is the verb spielen in the second position, and could another element come first?

German main clauses follow the verb-second (V2) rule: the finite verb must occupy the second slot. Here the subject Die Kinder is first, so spielen is second. You can front other elements (e.g. a time or place expression), but spielen still stays in second position, for example:
Auf dem Spielplatz spielen die Kinder zusammen.

What part of speech is zusammen, and where can it go in the sentence?

zusammen is an adverb of manner meaning “together.” Adverbs of manner typically follow the finite verb in German. In the example it comes right after spielen, but you can also move it to the end without changing the core meaning:
Die Kinder spielen auf dem Spielplatz zusammen.

Why is dem used in auf dem Spielplatz, and how do I know which case to use?

Spielplatz is masculine (der Spielplatz). The preposition auf is one of the two-way (Wechsel-) prepositions:

It takes dative when indicating a static location (answering “where?”).
It takes accusative when indicating movement toward something (answering “where to?”).
Here the children are already on the playground (location), so you use the dative masculine article dem: auf dem Spielplatz.

Could I say Die Kinder spielen zusammen im Spielplatz or drop the article (auf Spielplatz)?

No. German nouns in such contexts generally require an article. im (in dem) would imply “inside,” which doesn’t fit for a playground, and dropping the article (auf Spielplatz) is ungrammatical. The idiomatic expression for “on the playground” is always auf dem Spielplatz.

Could I use am Spielplatz instead of auf dem Spielplatz?

am is a contraction of an dem and also marks location. Grammatically it’s possible (Die Kinder spielen zusammen am Spielplatz), but German speakers usually say auf dem Spielplatz, since playgrounds are viewed as surfaces you play on rather than points you stand at.

Why is Spielplatz written as one word and capitalized?

German forms compound nouns by joining words: Spiel (play) + Platz (place) = Spielplatz (playground). All German nouns are always capitalized, whether simple or compound.

What’s the difference between zusammen and miteinander?

Both mean “together.” zusammen is more common and colloquial, while miteinander emphasizes mutual action and can sound a bit more formal or stylistic. In most contexts they’re interchangeable:
Die Kinder spielen zusammen.
Die Kinder spielen miteinander.