Vor der Untersuchung ziehe ich den Ärmel hoch, damit die Ärztin die Wunde sehen kann.

Because vor is a two-way preposition, and here it is used in a time expression meaning before the examination. In this kind of use, it takes the dative case.

• die Untersuchung = the examination
• dative singular feminine = der Untersuchung

So:

• vor der Untersuchung = before the examination

If vor described movement toward a place, you might see the accusative, but that is not what is happening here.

German main clauses usually put the finite verb in second position.

Here, Vor der Untersuchung comes first, so the verb must come next:

• Vor der Untersuchung = position 1
• ziehe = position 2
• ich = then the subject comes after the verb

So:

• Ich ziehe den Ärmel hoch.
• Vor der Untersuchung ziehe ich den Ärmel hoch.

Both are correct. When a time phrase is moved to the front, German uses this inversion automatically.

Yes. hochziehen is a separable verb.

Its base meaning is to pull up. In a normal main clause, the prefix hoch- moves to the end:

• Ich ziehe den Ärmel hoch.

But in the infinitive or in some other structures, it stays together:

• Ich möchte den Ärmel hochziehen.

So in your sentence:

• ziehe ... hoch = the separable verb hochziehen

Because Ärmel is the direct object of ziehen, and direct objects usually take the accusative case.

The noun is:

• der Ärmel = the sleeve

In the accusative singular masculine, der changes to den:

• nominative: der Ärmel
• accusative: den Ärmel

So:

• Ich ziehe den Ärmel hoch. = I pull up the sleeve.

German often uses the definite article with body parts and clothing when ownership is obvious from the context.

So instead of saying:

• I pull up my sleeve

German very naturally says:

• Ich ziehe den Ärmel hoch.

It still clearly means my sleeve, because the subject is ich and the action is happening to the speaker’s own clothing.

You can say meinen Ärmel if you want to emphasize that it is specifically my sleeve, but the version with den is very normal and idiomatic.

Ärmel means sleeve, as in the sleeve of a shirt, jacket, or coat.

So:

• den Ärmel hochziehen = to pull up the sleeve

This is a very common expression in medical contexts, for example before a blood test, vaccination, or examination.

Here damit means so that or in order that.

It introduces a purpose clause: it explains why the speaker pulls up the sleeve.

• Ich ziehe den Ärmel hoch, damit die Ärztin die Wunde sehen kann.
• I pull up the sleeve so that the doctor can see the wound.

So the relationship is:

• action: I pull up the sleeve
• purpose: so that the doctor can see the wound

Not naturally in this sentence, because um ... zu is usually used when the subject of both actions is the same.

For example:

• Ich lerne Deutsch, um in Deutschland zu arbeiten.
• The same person does both actions: I learn / I work

But here the subjects are different:

• ich pull up the sleeve
• die Ärztin sees the wound

Because the subjects are different, German normally uses damit:

• ..., damit die Ärztin die Wunde sehen kann.

That is the correct choice here.

Because damit introduces a subordinate clause, and in German subordinate clauses the finite verb goes to the end.

The normal main-clause order would be something like:

• Die Ärztin kann die Wunde sehen.

But after damit, the conjugated verb kann moves to the end:

• ..., damit die Ärztin die Wunde sehen kann.

Also notice that sehen stays near the end too, because it is paired with the modal verb kann.

Because kann is the finite modal verb, and in a subordinate clause it goes to the end.

With a modal verb, the other verb appears as an infinitive:

• sehen = to see
• kann = can

In a main clause:

• Die Ärztin kann die Wunde sehen.

In the damit clause:

• ..., damit die Ärztin die Wunde sehen kann.

So sehen kann is exactly what you expect in this structure.

Yes. die Ärztin means the female doctor.

German often marks professions by gender:

• der Arzt = male doctor
• die Ärztin = female doctor

So this sentence specifically tells you the doctor is female.

If the doctor were male, the sentence would say:

• ..., damit der Arzt die Wunde sehen kann.

Because Wunde is a feminine noun, and in the accusative singular, feminine articles do not change.

So:

• nominative: die Wunde
• accusative: die Wunde

That is why it looks the same here.

Compare that with masculine nouns, where the article usually changes:

• der Ärmel → den Ärmel

Untersuchung usually means examination, medical examination, or check-up, depending on context.

In this sentence, it most naturally refers to a medical examination by a doctor.

So:

• Vor der Untersuchung = Before the examination / Before the check-up

All of those are reasonable English meanings, but examination is the closest direct match.

Yes, it is natural and idiomatic.

A German speaker would understand it immediately as something like:

• Before the examination, I pull up my sleeve so that the doctor can see the wound.

Everything in it is standard and common:

• vor + dative for time
• verb-second in the main clause
• hochziehen as a separable verb
• damit for purpose
• verb-final in the subordinate clause

So this is a very good example sentence for everyday medical German.