Die Untersuchung beim Arzt dauert nur wenige Minuten.

Because Untersuchung is a feminine noun in German, so its nominative singular article is die.

• die Untersuchung = the examination / the check-up

German nouns have grammatical gender, and you usually have to learn the noun together with its article:

• der Arzt
• die Untersuchung
• das Krankenhaus

Here, Untersuchung is the subject of the sentence, so it appears in the nominative case: die Untersuchung.

Yes. Untersuchung comes from the verb untersuchen, which means to examine or to investigate.

A very common pattern in German is:

• verb: untersuchen = to examine
• noun: die Untersuchung = the examination

So if you know the verb, the noun is easier to remember.

Beim is a contraction of bei dem.

• bei dem Arzt → beim Arzt

This kind of contraction is very common in German and sounds more natural in everyday language.

Some other common ones are:

• in dem → im
• an dem → am
• zu dem → zum
• zu der → zur

So beim Arzt literally comes from bei dem Arzt.

Arzt is in the dative case because the preposition bei always takes the dative.

• bei + dative
• dem Arzt = dative singular of der Arzt

So:

• bei dem Arzt
• contracted: beim Arzt

This use of bei often means:

• at
• with
• at the place of

So beim Arzt can mean something like at the doctor’s, with the doctor, or at the doctor’s office, depending on context.

It can suggest either, depending on context.

In this sentence, beim Arzt most naturally means something like:

• at the doctor’s
• at the doctor’s office
• during a visit to the doctor

German often uses beim Arzt, beim Friseur, beim Bäcker, etc., where English may prefer:

• at the doctor’s
• at the hairdresser’s
• at the baker’s

So it is a very natural German expression.

Because German main clauses normally follow the verb-second rule.

In a normal statement, the finite verb goes in the second position:

• Die Untersuchung | dauert | nur wenige Minuten

Here:

1. Die Untersuchung = first position
2. dauert = second position

That is standard German word order.

Because the subject is die Untersuchung, which is third person singular.

The verb is dauern = to last. Its present tense forms are:

• ich dauere
• du dauerst
• er/sie/es dauert
• wir dauern
• ihr dauert
• sie/Sie dauern

Since die Untersuchung is grammatically like sie in the singular, the correct form is dauert.

Wenige here is being used as a determiner-like word meaning few.

• nur wenige Minuten = only a few minutes

No article is needed because wenige itself is already doing the job of describing the noun quantity.

Compare:

• nur wenige Minuten = only a few minutes
• nur ein paar Minuten = only a couple of minutes / only a few minutes
• die wenigen Minuten = the few minutes

So wenige without an article is completely normal here.

Because this phrase is in the accusative plural, and wenige here has the ending that matches that form.

The verb dauern often takes an expression of duration in the accusative:

• eine Minute dauern
• zwei Stunden dauern
• wenige Minuten dauern

So:

• wenige Minuten = accusative plural

You would see wenigen Minuten in a dative plural context, for example:

• nach wenigen Minuten = after a few minutes

That is why the sentence has wenige Minuten, not wenigen Minuten.

Because wenige means few, so the noun must be plural.

• eine Minute = one minute
• wenige Minuten = a few minutes

Also, Minute is a feminine noun:

• singular: die Minute
• plural: die Minuten

So in this sentence, the plural form Minuten is required.

Yes, nur can sometimes move, but its position affects emphasis.

In Die Untersuchung beim Arzt dauert nur wenige Minuten, nur directly limits wenige Minuten:

• it lasts only a few minutes

That is the most natural position here.

If you move nur, the focus can shift. For example:

• Nur die Untersuchung beim Arzt dauert wenige Minuten
  = only the examination at the doctor’s lasts a few minutes

So the original placement is best if you want to say the duration is short.

Yes, absolutely.

That version is also grammatical:

• Beim Arzt dauert die Untersuchung nur wenige Minuten.

When Beim Arzt is moved to the front, it takes the first position, and the verb still stays in second position:

• Beim Arzt | dauert | die Untersuchung | nur wenige Minuten

The meaning is very similar, but the emphasis changes a little:

• Die Untersuchung beim Arzt... focuses first on the examination
• Beim Arzt... focuses first on the doctor/doctor’s visit context

Both are natural German.