Die Kinder springen im Garten und lachen.

Yes—die can be feminine singular, but it is also the definite article for plural nouns in the nominative and accusative.

Here, die Kinder means the children.

So in this sentence:

• das Kind = the child
• die Kinder = the children

The die here is plural, not feminine singular.

Because Kinder is the nominative plural form, and it is the subject of the sentence—the ones doing the actions.

• die Kinder = the children → subject
• den Kindern = to the children / the children (in dative)

Since the children are the ones jumping and laughing, German uses the nominative:

• Die Kinder springen ...

Because the subject is die Kinder, which is third-person plural (they).

The verb springen is conjugated like this in the present tense:

• ich springe
• du springst
• er/sie/es springt
• wir springen
• ihr springt
• sie springen

Since die Kinder = they, the correct form is springen.

Because German, like English, usually does not repeat the subject when the same subject does two actions connected by und.

So:

• Die Kinder springen im Garten und lachen.

means:

• The children jump in the garden and laugh.

You could repeat the subject, but it would usually sound unnecessary here:

• Die Kinder springen im Garten und die Kinder lachen.

That is grammatical, but not natural in normal speech.

im is a contraction of:

• in dem → im

So:

• im Garten = in the garden

This is very common in German.

Other common contractions include:

• am = an dem
• zum = zu dem
• zur = zu der

Because this sentence describes location, not movement toward a destination.

German often makes this distinction with prepositions like in:

• in + dative = location, in / inside
• in + accusative = direction, into

So:

• im Garten = in the garden → location
• in den Garten = into the garden → movement into it

Here the children are already doing the action in the garden, so German uses the dative:

• im Garten

Because the preposition in can take either the dative or the accusative, depending on meaning.

Here it answers the question where?

• Where are they jumping and laughing?
• Im Garten

When in means location, it takes the dative.

Since Garten is masculine:

• nominative: der Garten
• dative: dem Garten

And in dem Garten becomes im Garten.

It can mean either, depending on context.

German present tense often covers both:

• Die Kinder springen im Garten.
  • The children jump in the garden.
  • The children are jumping in the garden.

English often prefers the progressive (are jumping) for something happening right now, but German usually just uses the normal present tense.

German main clauses usually follow the pattern:

• subject + verb + other information

So here:

• Die Kinder = subject
• springen = verb
• im Garten = prepositional phrase
• und lachen = second coordinated verb/action

The sentence is straightforward because it is a normal main clause with no inversion.

You could also move im Garten to the front for emphasis:

• Im Garten springen die Kinder und lachen.

That is grammatical, but it sounds more marked and emphasizes where the action happens.

Because in German, a comma is usually not used before und when it simply connects words, phrases, or clauses in a basic way.

Here und connects two actions with the same subject:

• springen
• lachen

So no comma is needed:

• Die Kinder springen im Garten und lachen.

Yes. Even though the subject is not repeated, lachen still agrees with die Kinder.

The full idea is:

• Die Kinder springen im Garten und [die Kinder] lachen.

Since the subject is plural, the correct verb form is:

• lachen

If the subject were singular, it would change:

• Das Kind springt im Garten und lacht.

Not exactly. Springen specifically means to jump or to leap.

So the core meaning is:

• The children are jumping in the garden and laughing.

A looser translation might sometimes sound natural in English depending on context, but if you want to stay close to the German, jumping is the best choice.