Я хочу ещё кусочек арбуза.

Because кусочек арбуза means a little piece of watermelon or a piece of watermelon.

In Russian, after nouns that indicate a part, amount, or piece of something, the thing itself is often put in the genitive case:

• кусочек арбуза = a piece of watermelon
• чашка чая = a cup of tea
• бутылка воды = a bottle of water

So арбуз becomes арбуза because it depends on кусочек.

Кусочек is the diminutive form of кусок.

• кусок = a piece, a chunk
• кусочек = a little piece, a small piece

Russian uses diminutives very often, not only to mean physical smallness, but also to sound:

• softer
• friendlier
• more natural in everyday speech

So Я хочу ещё кусочек арбуза sounds very natural, like I want another little piece of watermelon or simply I want some more watermelon in a friendly everyday way.

Here ещё means more, another, or some more.

So:

• Я хочу кусочек арбуза = I want a piece of watermelon.
• Я хочу ещё кусочек арбуза = I want another piece of watermelon / I want some more watermelon.

Be careful: ещё can also mean still in other sentences.

For example:

• Он ещё дома. = He is still at home.

So the meaning of ещё depends on context.

Кусочек is in the accusative singular, because it is the direct object of хочу.

The verb хотеть usually takes an object in the accusative:

• Я хочу чай. = I want tea.
• Я хочу яблоко. = I want an apple.
• Я хочу кусочек арбуза. = I want a piece of watermelon.

For inanimate masculine nouns, the accusative is often the same as the nominative.
So:

• nominative: кусочек
• accusative: кусочек

That is why the form does not visibly change.

Yes, absolutely.

• Я хочу ещё кусочек арбуза = I want another/some more piece of watermelon.
• Я хочу ещё один кусочек арбуза = I want one more piece of watermelon.

Adding один makes the idea of one more more explicit.

Without один, the sentence is still perfectly natural and often sounds smoother in conversation.

Yes, you can very naturally say:

• Хочу ещё кусочек арбуза.

Russian often drops subject pronouns when the meaning is clear from the verb ending.

Since хочу already means I want, я is optional.

Compare:

• Я хочу ещё кусочек арбуза. = slightly more explicit
• Хочу ещё кусочек арбуза. = very natural in conversation

Both are correct.

Yes. Russian word order is flexible, although some orders sound more neutral than others.

The most neutral version is:

• Я хочу ещё кусочек арбуза.

But you might also hear:

• Ещё кусочек арбуза я хочу.
• Я ещё хочу кусочек арбуза.

These can shift emphasis a little.

For example:

• Я хочу ещё кусочек арбуза. = neutral
• Я ещё хочу кусочек арбуза. = emphasis on still / also want
• Ещё кусочек арбуза! = more like an exclamation, such as at the table

So yes, the words can move, but the neutral learner-friendly order is the original one.

Russian often does not need a separate word for some where English does.

In this sentence, the meaning of some/a piece of is already expressed by:

• кусочек = a little piece
• plus the genitive арбуза

So кусочек арбуза naturally gives the idea of a piece of watermelon or some watermelon.

Russian often leaves this kind of meaning implicit instead of adding a separate word.

The stress is:

• Я хочУ ещё кусОчек арбУза.

A rough pronunciation guide:

• я = ya
• хочу = kha-CHOO
• ещё = yi-SHCHO
• кусочек = koo-SO-chek
• арбуза = ar-BOO-za

A few notes:

• ё is always stressed
• х is like the ch in Scottish loch or German Bach
• ч is like English ch
• я at the beginning is like ya

Both can mean that you want more watermelon, but they are not exactly the same.

• ещё кусочек арбуза = another little piece of watermelon
• ещё арбуза = some more watermelon

The first one focuses on a piece as a unit.
The second one is more about more of the substance/food in general.

So if watermelon is being served in pieces, ещё кусочек арбуза is especially natural.

Yes, it sounds quite natural and pleasant, especially because кусочек is soft and friendly.

If you want to make it more clearly polite, you could say:

• Можно мне ещё кусочек арбуза? = May I have another piece of watermelon?
• Я бы хотел ещё кусочек арбуза. = I would like another piece of watermelon.
• Дайте, пожалуйста, ещё кусочек арбуза. = Please give me another piece of watermelon.

But the original sentence is completely normal if you are simply saying what you want.