Das Kind spielt mit der Puppe.

Because Kind is a neuter noun in German. Its dictionary form is das Kind. In this sentence, das Kind is the subject, so it is in the nominative case, and the nominative singular article for a neuter noun is das.

A useful thing to remember: German noun gender is grammatical, so Kind is neuter even though a real child is of course not an it in the English sense.

Spielen is the infinitive, meaning to play.
In the sentence, the subject is das Kind, which is third person singular, so the verb changes to spielt.

Present tense of spielen:

• ich spiele
• du spielst
• er / sie / es spielt
• wir spielen
• ihr spielt
• sie / Sie spielen

So das Kind spielt = the child plays / is playing.

Because mit always takes the dative case.

The noun Puppe is feminine, so:

• nominative: die Puppe
• dative: der Puppe

So after mit, you must say mit der Puppe, not mit die Puppe.

It is dative singular.

You can tell this because the preposition mit requires the dative. So the phrase breaks down like this:

• mit = preposition
• der = feminine dative singular article
• Puppe = noun

So mit der Puppe is a dative prepositional phrase.

Here, mit means with and introduces the thing the child is playing with.

So:

• Das Kind spielt. = The child is playing.
• Das Kind spielt mit der Puppe. = The child is playing with the doll.

In other words, mit der Puppe adds extra information about what the play involves.

It can mean either one, depending on context.

German uses the simple present much more broadly than English does. So:

• Das Kind spielt mit der Puppe can mean The child plays with the doll
• or The child is playing with the doll

If you need to know which one is meant, you usually rely on context.

Because all nouns are capitalized in German.

So in this sentence:

• Kind is capitalized because it is a noun
• Puppe is capitalized because it is a noun

This is a standard rule in German writing.

In a normal German main clause, the finite verb usually comes in second position.

Here:

• position 1: Das Kind
• position 2: spielt
• then: mit der Puppe

So the pattern is:

subject + finite verb + other information

That is one of the most important word-order rules in German.

Yes. German word order is flexible, but the finite verb still has to stay in second position in a main clause.

For example:

• Mit der Puppe spielt das Kind.

This is still grammatically correct. It puts more emphasis on mit der Puppe.

What you cannot do in a normal main clause is move spielt away from second position.

A pronoun version could be:

Es spielt mit ihr.

Why?

• das Kind is grammatically neuter, so it can become es
• die Puppe becomes ihr after mit, because mit takes the dative

So:

• das Kind → es
• mit der Puppe → mit ihr

In real life, if the child is known to be a boy or girl, speakers may also use er or sie for the child, depending on context.