Nach dem Regen springt mein Neffe in jede Pfütze und lacht laut.

Because nach normally takes the dative case in expressions like this.

• der Regen = the rain
• Dative singular becomes dem Regen

So:

• nach dem Regen = after the rain

This is something you often just learn together with the preposition:

• nach + dative

German main clauses follow the verb-second rule: the finite verb must come in the second position.

Here, the sentence starts with the time phrase:

• Nach dem Regen = first position

So the verb must come next:

• springt = second position

Then the subject comes after it:

• mein Neffe

So the structure is:

• Nach dem Regen | springt | mein Neffe ...

If you started with the subject instead, you would get:

• Mein Neffe springt nach dem Regen in jede Pfütze und lacht laut.

Both are correct.

Because Neffe is masculine singular nominative, and possessive words like mein behave a bit like ein.

For masculine nominative singular, the form is simply:

• mein Neffe
• dein Bruder
• sein Vater

So mein is the correct form here because mein Neffe is the subject of the sentence.

Because springt is the 3rd person singular present tense form of springen.

The subject is mein Neffe = my nephew, which is he.

So:

• springen = infinitive, to jump
• er springt = he jumps

Likewise:

• lachen = to laugh
• er lacht = he laughs

Here in is followed by the accusative because it expresses movement toward/into something.

Your nephew is not already in the puddle; he is jumping into it.

So:

• in + accusative = direction / movement
• in + dative = location / being inside

Examples:

• Er springt in die Pfütze. = He jumps into the puddle.
• Er steht in der Pfütze. = He is standing in the puddle.

In your sentence:

• Pfütze is feminine
• feminine accusative singular of jede is jede

So in jede Pfütze is correct.

They do, but in this particular form, feminine accusative singular happens to look the same as feminine nominative singular.

So:

• jede Pfütze = nominative feminine singular
• in jede Pfütze = accusative feminine singular

They look the same, even though the case is different.

Compare with masculine, where you can see the change more clearly:

• jeder Park = nominative
• in jeden Park = accusative

So the lack of visible change here is just because of the feminine form.

Jede means every or each.

So in jede Pfütze springen suggests that he jumps into every puddle he comes across, one after another.

It gives the sentence a playful feeling, as if he cannot resist puddles after it rains.

Because the subject is the same in both parts of the sentence, and German often leaves it out in coordinated clauses when it is obvious.

So this:

• Nach dem Regen springt mein Neffe in jede Pfütze und lacht laut.

means the same as the more repetitive version:

• Nach dem Regen springt mein Neffe in jede Pfütze und mein Neffe lacht laut.

The repeated subject is unnecessary, so it is omitted.

German usually uses the same form for adjectives and adverbs.

So:

• laut can mean loud or loudly, depending on context

Here it functions adverbially:

• lacht laut = laughs loudly

This is very normal in German.

Other examples:

• Er spricht leise. = He speaks quietly.
• Sie singt schön. = She sings beautifully.

No special -ly ending is needed.

In this sentence, it is a time expression meaning after the rain.

It tells you when the action happens.

So the idea is:

• first it rains
• afterward, the nephew jumps into puddles and laughs

In other contexts, nach can also mean things like to, toward, or according to, but here it is clearly temporal.

Yes. German present tense can describe:

• something happening right now
• something that happens regularly or typically

So this sentence could mean either:

• a scene happening now, or
• a general habit: after it rains, he always jumps in puddles and laughs

German present tense is flexible in that way, much like English He jumps in every puddle.

Because all nouns are capitalized in German.

So in this sentence:

• Regen = noun
• Neffe = noun
• Pfütze = noun

This is one of the most noticeable spelling differences between German and English.

A helpful rough guide is:

• Pfütze ≈ PFYT-tsuh

A few parts may feel unusual for English speakers:

• pf is pronounced together at the start
• ü is a front rounded vowel; a rough approximation is to say ee while rounding your lips
• tz sounds like ts

So it may take a little practice, but that is completely normal for learners.