Мы взяли по куску дыни, а детям дали по куску арбуза.

Here по has a distributive meaning: one each / apiece / per person.

So:

• Мы взяли по куску дыни = We each took a piece of melon
• детям дали по куску арбуза = the children were given a piece of watermelon each

This is a very common Russian pattern:

• по яблоку = an apple each
• по билету = a ticket each
• по чашке чая = a cup of tea each

Because after по in this distributive sense, Russian commonly uses the dative singular.

So:

• кусок = nominative
• куску = dative singular

That is why you get по куску.

This pattern is very common:

• по яблоку
• по подарку
• по вопросу
• по куску

Even though several people are involved, the thing distributed is still one item per person, so Russian uses the singular form.

Both are in the genitive singular.

• дыня → дыни
• арбуз → арбуза

They are genitive because they depend on кусок:

• кусок дыни = a piece of melon
• кусок арбуза = a piece of watermelon

This is the normal pattern after words like:

• кусок хлеба = a piece of bread
• стакан воды = a glass of water
• чашка чая = a cup of tea

So дыни and арбуза are not the main object directly; they are part of the phrase piece of melon / piece of watermelon.

Because дать takes the recipient in the dative case.

Ask the question:

• дать кому? = give to whom?

Answer:

• детям = to the children

So:

• дали детям = gave to the children

This is a basic Russian pattern:

• дать другу книгу = give a friend a book
• сказать маме = tell mom
• помочь брату = help a brother

Russian often omits a repeated subject when it is clear from context.

The first clause has мы:

• Мы взяли...

In the second clause, Russian does not need to repeat мы:

• а детям дали...

So the natural understanding is:

• We took a piece of melon each, and gave the children a piece of watermelon each

If you wanted, you could say а детям мы дали..., but it is not necessary.

Here а is a contrastive connector. It often means something like:

• and
• while
• whereas
• as for

It does not always mean a strong but.

In this sentence, it contrasts two groups:

• мы got melon
• детям got watermelon

So а helps set up that contrast: We took melon, while the children got watermelon.

They are perfective because the sentence describes completed actions in the past.

• взяли = took as a completed act
• дали = gave as a completed act

This fits the meaning well: the pieces were taken and given once, as finished events.

If you used imperfective verbs:

• брали
• давали

the meaning would usually shift toward something like:

• repeated action
• ongoing action
• habitual action
• focus on the process rather than the result

So in this sentence, взяли and дали are the natural choice.

Yes, the normal meaning is one piece each.

So:

• Мы взяли по куску дыни = each of us took one piece of melon
• детям дали по куску арбуза = each child got one piece of watermelon

If the speaker wanted to say several pieces each, Russian would usually express that differently, for example with a number:

• по два куска = two pieces each
• по три куска = three pieces each

Yes. Russian word order is fairly flexible, but different orders sound more or less natural depending on what is being emphasized.

The given sentence is very natural because it sets up a contrast:

• Мы ...
• а детям ...

That makes the structure easy to follow.

You might also see things like:

• Мы взяли по куску дыни, а детям мы дали по куску арбуза.
• Мы по куску дыни взяли, а детям по куску арбуза дали.

These are possible, but the original version is the most neutral and smooth.

Because the direct object is not simply melon or watermelon. The object is the whole phrase:

• по куску дыни
• по куску арбуза

The key noun is кусок. Then дыни / арбуза depend on it in the genitive.

So Russian is not saying:

• took melon
• gave watermelon

It is saying:

• took a piece of melon
• gave a piece of watermelon

That is why the fruit words are not in the accusative.