Für das Frühstück morgen brauchst du die Melone nicht zu schneiden; ich mache das schon.

Because German main clauses normally put the finite verb in second position.

Here, Für das Frühstück morgen takes the first position, so brauchst must come next, and then the subject du follows:

• Für das Frühstück morgen | brauchst | du | ...

If you started with the subject instead, that would also be correct:

• Du brauchst die Melone für das Frühstück morgen nicht zu schneiden.

German is flexible about what comes first, but the finite verb still stays in second position.

This is a common German pattern:

• brauchen + object + nicht + zu + infinitive

It means to not need to do something.

So:

• Du brauchst die Melone nicht zu schneiden
  = You don’t need to cut the melon

In this use, brauchen behaves a bit like a modal verb, but it is most common in negative contexts such as:

• nicht
• nie
• kaum

So learners often meet it as nicht brauchen zu + infinitive.

Because brauchen in this pattern takes a zu-infinitive.

Compare:

• Du musst die Melone nicht schneiden.
• Du brauchst die Melone nicht zu schneiden.

Both can mean You don’t need to cut the melon, but only brauchen uses zu here.

That is one reason why brauchen is not a fully normal modal verb like müssen, können, or dürfen.

Usually not in normal everyday German.

A sentence like:

• Ich brauche die Melone zu schneiden

sounds odd or very unnatural to most speakers.

German usually prefers:

• Ich muss die Melone schneiden.
  = I have to cut the melon

or regular brauchen with a noun:

• Ich brauche ein Messer.
  = I need a knife

So the brauchen + zu-infinitive pattern is most natural when there is some kind of negation, as in your sentence.

Because nicht is negating the action zu schneiden.

The meaning is:

• there is no need to do the cutting

So nicht belongs with the infinitive phrase.

If you moved nicht earlier, the emphasis could change. For example:

• Du brauchst nicht die Melone zu schneiden

can sound like it’s not the melon that needs cutting or not the melon specifically.

So in your sentence, the placement of nicht clearly gives the meaning you do not need to cut the melon.

It means you don’t need to cut the melon.

That is an important difference.

• Du brauchst die Melone nicht zu schneiden
  = no necessity
• Du darfst die Melone nicht schneiden
  = you must not / you are not allowed to cut it
• Schneide die Melone nicht
  = don’t cut the melon

So nicht brauchen zu is about lack of necessity, not a prohibition.

Both are in the accusative, but for different reasons.

• für das Frühstück: the preposition für always takes the accusative
• die Melone: this is the direct object of schneiden, so it is accusative too

A detail that can be confusing:

• das Frühstück is neuter singular, and its accusative article is still das
• die Melone is feminine singular, and its accusative article is also die

So even though both are accusative, the articles do not visibly change here.

Here it means something like:

• I’ll do it
• I’ll take care of it
• Leave it to me

It is a very natural, reassuring kind of sentence.

The word schon here does not mainly mean already. Instead, it adds a tone of reassurance or confidence:

• Don’t worry, I’ll handle it.

So ich mache das schon is often less about time and more about attitude.

Because das often refers back to a whole action, idea, or situation.

Here, das refers to the task just mentioned:

• cutting the melon

German very often uses das when pointing back to an entire previous statement or action.

So:

• Ich mache das schon

sounds very natural here.

Es is not impossible in every context, but das is the more idiomatic choice when referring to the whole task.

Yes, it is natural and grammatical.

A very common shortened form is:

• Fürs Frühstück morgen brauchst du die Melone nicht zu schneiden; ich mache das schon.

because für das often becomes fürs.

A few notes:

• für das Frühstück morgen = for tomorrow’s breakfast
• the wording is fine as it is
• in everyday speech, some people might also choose slightly different phrasing, but your sentence sounds normal

Also, the semicolon is fine here because it connects two complete main clauses. A full stop would also work:

• Für das Frühstück morgen brauchst du die Melone nicht zu schneiden. Ich mache das schon.

A simple comma would not be standard here unless the sentence were rewritten.