Mein Bruder sucht das Bügeleisen, weil er sein Hemd noch schnell bügeln will.

Because weil introduces a subordinate clause.

In German, subordinate clauses are normally separated from the main clause by a comma:

• Mein Bruder sucht das Bügeleisen, weil ...
• main clause, subordinate clause

So the comma is not optional here.

Because weil sends the conjugated verb to the end of its clause.

In the main clause, German normally has the finite verb in second position:

• Mein Bruder sucht das Bügeleisen.

But after weil, the clause becomes subordinate, so the finite verb moves to the end:

• weil er ... will

That is one of the most important German word-order rules.

In a subordinate clause with a modal verb, the main infinitive comes before the conjugated modal at the end:

• weil er sein Hemd noch schnell bügeln will

Compare that with the main-clause order:

• Er will sein Hemd noch schnell bügeln.

So:

• main clause: finite verb earlier → will ... bügeln
• subordinate clause: finite verb at the end → bügeln will

Noch schnell is a very common expression.

Here:

• noch = still / before it is too late / one more thing before something else
• schnell = quickly

Together, noch schnell often means something like:

• quickly
• just quickly
• before anything else, quickly

So the feeling is that he wants to iron his shirt quickly right now, probably before leaving or before doing something else.

Here sein means his, and it refers to er, which refers back to mein Bruder.

So:

• mein Bruder = my brother
• er = he
• sein Hemd = his shirt

A common point of confusion: the form sein does not mean the shirt is neuter because of the owner. It means the owner is he.

The form changes according to the noun that follows:

• sein Hemd = his shirt
• seine Jacke = his jacket
• seinen Mantel = his coat

They are both direct objects, so they are in the accusative case.

• suchen takes an accusative object
  → das Bügeleisen
• bügeln also takes an accusative object
  → sein Hemd

A useful detail: with neuter singular nouns, nominative and accusative often look the same.

So:

• das Bügeleisen could be nominative or accusative in form
• das Hemd also looks the same in nominative and accusative

That is why the case is determined by the verb’s role in the sentence, not by a visible ending here.

No. German normally requires an explicit subject.

So even though mein Bruder was already mentioned, the subordinate clause still needs its own subject:

• weil er sein Hemd ... bügeln will

Unlike in some languages, German does not usually drop subject pronouns just because the meaning is clear.

Because the sentence describes:

• what he is doing now: Er sucht das Bügeleisen
• why he is doing it: because he wants to iron his shirt

German very often uses the present tense and modal verbs for near-future meaning.

So bügeln will does not mean a distant future. It means something like:

• he wants to iron it
• he intends to iron it
• he is about to iron it

That is perfectly natural German.

Yes. Both are possible:

• das Bügeleisen suchen
• nach dem Bügeleisen suchen

The version without nach is very common and direct:

• Mein Bruder sucht das Bügeleisen.

The version with nach can sound a bit more like searching for:

• Mein Bruder sucht nach dem Bügeleisen.

If you use nach, the noun changes to the dative:

• nach dem Bügeleisen

So:

• sucht das Bügeleisen = accusative object
• sucht nach dem Bügeleisen = prepositional phrase with dative

Because they are nouns, and all nouns are capitalized in German.

So in this sentence:

• Bruder
• Bügeleisen
• Hemd

are capitalized simply because they are nouns.

This is a basic and very consistent rule in German spelling.

It is a compound noun, which is extremely common in German.

It is made from:

• bügeln = to iron
• Eisen = iron

So Bügeleisen literally means something like ironing iron, but in normal English it is simply iron or clothes iron.

German often builds vocabulary this way by combining smaller words into one larger noun.

Yes, absolutely.

You can say:

• Weil er sein Hemd noch schnell bügeln will, sucht mein Bruder das Bügeleisen.

That is also correct.

But when the subordinate clause comes first, the main clause still follows the normal rule that the finite verb comes early, so you get:

• ..., sucht mein Bruder ...

not:

• ..., mein Bruder sucht ... in the sense of an extra first element before sucht

So the original sentence and the reversed version are both grammatical; the difference is mainly emphasis.