Ich brauche heute keinen Rasierer, aber mein Bruder sucht seinen schon wieder.

Because Rasierer is the direct object of brauchen, so it must be in the accusative case.

Rasierer is masculine singular, and the accusative form of kein for a masculine noun is keinen.

A quick comparison:

• nominative: kein Rasierer
• accusative: keinen Rasierer

So:

• Ein Rasierer ist da.
• Ich brauche keinen Rasierer.

Kein is used to negate a noun that would otherwise have an indefinite article.

So:

• ein Rasierer = a razor
• kein Rasierer = no razor / not a razor / no razor at all

In this sentence, keinen Rasierer means no razor / not any razor.

English speakers often want to use nicht, but German usually prefers:

• Ich habe keinen Rasierer. not
• Ich habe nicht einen Rasierer.

That second version is possible only in special emphatic contexts, not as the normal neutral wording.

Seinen is also in the accusative masculine singular.

The full idea is:

• mein Bruder sucht seinen Rasierer

Here, Rasierer is again the direct object of sucht, so it is accusative. Since Rasierer is masculine singular, the possessive form must match that:

• nominative masculine: sein Rasierer
• accusative masculine: seinen Rasierer

In the actual sentence, Rasierer is left out because it is already understood, but seinen keeps the same form.

Because German often leaves out a noun when it is obvious from context.

So:

• mein Bruder sucht seinen Rasierer schon wieder can become
• mein Bruder sucht seinen schon wieder

Here seinen means his one / his razor, with Rasierer understood.

English does something similar in sentences like:

• I’ll take the red one.
• He can’t find his.

German simply allows the same kind of omission here.

Both, in different ways:

• sein- tells you whose it is: it belongs to my brother
• -en shows the grammatical form needed for the omitted noun Rasierer

So the possessive base comes from the owner:

• mein Bruder = third-person singular masculine
• therefore: sein-

But the ending is determined by the thing possessed:

• Rasierer = masculine singular accusative
• therefore: seinen

This is a very common pattern in German possessives.

Because aber is a coordinating conjunction, not a subordinating one.

After aber, German starts another main clause, so the conjugated verb still stays in second position:

• ..., aber mein Bruder sucht ...

Compare that with a subordinating conjunction like weil, which sends the verb to the end:

• ..., weil mein Bruder seinen schon wieder sucht.

So:

• aber = normal main-clause word order
• weil = verb goes to the end

Because the sentence joins two main clauses:

• Ich brauche heute keinen Rasierer
• aber mein Bruder sucht seinen schon wieder

In German, when two main clauses are linked by conjunctions like aber, a comma is normally used.

So the comma is not optional here in standard German spelling.

In a main clause, the conjugated verb is in position 2, and time expressions like heute often appear early in the middle field.

So this is a very normal order:

• Ich brauche heute keinen Rasierer.

You could also move things around for emphasis, for example:

• Heute brauche ich keinen Rasierer.

That version puts more emphasis on today. The original sentence is the more neutral flow.

Schon wieder means again, often with the feeling of yet again or again already.

It often suggests repetition that the speaker notices, and sometimes mild annoyance or amusement.

So in this sentence, it suggests that the brother has been looking for his razor repeatedly, or that this is a familiar situation.

Compare:

• wieder = again
• schon wieder = again already / yet again

The second one often sounds more expressive.

German word order in the middle part of the clause is flexible, and adverbial expressions like schon wieder can move depending on emphasis.

Here:

• mein Bruder sucht seinen schon wieder

sounds natural and puts seinen before schon wieder, with the repeated action coming last.

You could also hear or say:

• mein Bruder sucht schon wieder seinen Rasierer

That version is also natural, especially when the noun is stated explicitly.

So the placement is not random, but it is somewhat flexible. The given sentence is a normal, idiomatic choice.

Yes. That is completely grammatical.

The version with seinen alone avoids repeating Rasierer, since the noun was already mentioned earlier:

• Ich brauche heute keinen Rasierer, aber mein Bruder sucht seinen schon wieder.

German often prefers this kind of omission when the meaning is obvious, because it sounds less repetitive.

So both are correct:

• ... sucht seinen Rasierer schon wieder
• ... sucht seinen schon wieder

The second one is simply more compact.

No. Here brauchen is just a normal full verb meaning to need, followed by a direct object:

• Ich brauche keinen Rasierer.

English speakers sometimes learn brauchen in structures like:

• Du brauchst nicht zu kommen.

That is a different pattern, where brauchen is used with zu + infinitive.

But in your sentence, it is the ordinary noun-object use:

• brauchen + accusative noun