Wenn der Mülleimer voll ist, brauchst du den Flur nicht sofort zu fegen; bring erst den Müll raus.

Because wenn introduces a subordinate clause, and in German subordinate clauses the conjugated verb usually goes to the end.

So:

• der Mülleimer ist voll = the trash can is full
• wenn der Mülleimer voll ist = when the trash can is full

This is one of the most important German word-order patterns to learn:

• main clause: verb in position 2
• subordinate clause: verb at the end

That is because of grammatical case.

• der Mülleimer is the subject of ist, so it is in the nominative case.
• den Flur is the direct object of fegen, so it is in the accusative case.
• den Müll is the direct object of rausbringen, so it is also in the accusative case.

So the article changes depending on the role of the noun in the sentence.

Because voll is being used as a predicate adjective, not directly before a noun.

Compare:

• der volle Mülleimer = the full trash can
  
  • here voll comes before the noun, so it gets an ending
• der Mülleimer ist voll = the trash can is full
  
  • here voll comes after ist, so it does not get an ending

This is similar to English: the full trash can vs the trash can is full.

This uses brauchen + nicht + zu + infinitive, which means to not need to do something.

So:

• du brauchst den Flur nicht sofort zu fegen
  = you do not need to sweep the hallway right away

The structure is:

• brauchen = to need
• nicht = not
• zu fegen = to sweep

A native English speaker may expect something like you need not sweep, and this German structure works in a similar way.

Because after brauchen in this meaning, German normally uses zu + infinitive.

So:

• Du brauchst nicht zu kommen. = You don’t need to come.
• Du brauchst den Flur nicht zu fegen. = You don’t need to sweep the hallway.

In informal spoken German, some people may omit zu, but learners should treat brauchen + nicht + zu + infinitive as the standard pattern.

Because the sentence starts with a wenn-clause, and when a subordinate clause comes first, the main clause follows with the verb in second position.

Structure:

• Wenn der Mülleimer voll ist,
• brauchst du den Flur nicht sofort zu fegen

German does not say:

• Wenn der Mülleimer voll ist, du brauchst ...

Instead, after the opening clause, the main clause begins and the verb comes first:

• ..., brauchst du ...

This is a very common word-order pattern.

• der Mülleimer = the trash can / garbage bin
• der Müll = the trash / garbage itself

So in this sentence:

• Wenn der Mülleimer voll ist = when the trash can is full
• bring erst den Müll raus = first take the trash out

One is the container; the other is what is inside it.

Because the verb is rausbringen, which is a separable verb.

Its parts are:

• bringen = to bring
• raus = out

In a main clause, separable prefixes usually move to the end:

• Ich bringe den Müll raus.
• Bring den Müll raus.

But in the infinitive, the verb is written together:

• den Müll rausbringen

This is a very common German pattern:

• aufstehen → Ich stehe auf
• einkaufen → Wir kaufen ein
• rausbringen → Sie bringt den Müll raus

Because this is the informal singular imperative—used when speaking to one person with du.

From bringen, the du-imperative is:

• bring!

So:

• Bring den Müll raus! = Take the trash out!

German imperatives often drop the -st ending and the pronoun du:

• du bringst → bring!

Some verbs also optionally keep an -e in formal or older-style language, but bring is the normal modern form here.

Here erst means first or for now, first do this before anything else.

So:

• bring erst den Müll raus
  = first take the trash out

It suggests order:

1. take out the trash
2. then worry about sweeping the hallway

Be careful: erst can also mean only / not until in other contexts, so its meaning depends on the sentence.

Raus is a shorter, very common spoken and everyday form of heraus.

So these are closely related:

• den Müll rausbringen
• den Müll herausbringen

In everyday German, raus is extremely common and natural. Heraus can sound a bit more formal, emphatic, or written depending on context.

The semicolon links two closely related parts:

• Wenn der Mülleimer voll ist, brauchst du den Flur nicht sofort zu fegen
• bring erst den Müll raus

A semicolon is stronger than a comma but not as final as a period. It suggests: these two ideas belong together.

In normal writing, a period would also often be possible:

• Wenn der Mülleimer voll ist, brauchst du den Flur nicht sofort zu fegen. Bring erst den Müll raus.

So the semicolon is mainly a punctuation choice, not a grammar rule that changes the meaning much.

Yes, but the nuance is a little different.

• Du brauchst den Flur nicht sofort zu fegen
  = you do not need to sweep the hallway right away
• Du musst den Flur nicht sofort fegen
  = you don’t have to sweep the hallway right away

In many situations, these are very close in meaning. But brauchen + nicht + zu + infinitive often sounds especially clear for there is no need to.

For learners, both are useful, but the sentence you have is a very natural way to express no need.

Because adverbs like sofort often appear before the infinitive phrase they modify.

Here:

• nicht sofort zu fegen = not to sweep right away

It is telling you when the sweeping does not need to happen.

Other placements are possible in German depending on emphasis, but this version is natural and clear. The important idea is that sofort belongs with the action fegen and means immediately / right away.