Lægen vil undersøge hendes håndled igen i morgen.

Because Danish usually adds the definite article to the end of the noun.

• en læge = a doctor
• lægen = the doctor

So lægen is the definite form of læge.

Vil is a modal verb. It is followed by the infinitive of the main verb, here undersøge.

So the structure is:

• lægen = subject
• vil = finite verb / modal
• undersøge = main verb in infinitive

This is very common in Danish, just like English will examine.

After modal verbs such as vil, kan, skal, and må, Danish uses the bare infinitive, without at.

So:

• Lægen vil undersøge ... = correct
• Lægen vil at undersøge ... = incorrect

Compare that with a verb that does require at:

• Lægen prøver at undersøge ...

It can mean both, depending on context.

• Jeg vil spise can mean I want to eat
• Det vil regne i morgen means It will rain tomorrow

In Lægen vil undersøge hendes håndled igen i morgen, vil most naturally expresses a future or intended action: the doctor is going to examine her wrist again tomorrow.

Yes. Danish often uses the present tense for future meaning when a time expression makes the future clear.

So you could also say:

• Lægen undersøger hendes håndled igen i morgen

That still refers to the future because i morgen makes the time clear. Using vil can add a sense of intention, plan, or prediction.

Because sin/sit/sine is only used when the possessor is the same as the subject.

Here the subject is lægen. The wrist belongs to her, not to the doctor, so Danish uses hendes.

• Lægen vil undersøge hendes håndled = the doctor will examine her wrist
• Lægen vil undersøge sit håndled = the doctor will examine his/her own wrist

Also, if you did use the reflexive form here, it would be sit, not sin, because håndled is a neuter noun.

Because a possessive like hendes already determines the noun.

Just like in English, you say:

• her wrist not
• her the wrist

So hendes håndled is the normal Danish structure.

Håndled is a neuter noun:

• et håndled = a wrist
• håndleddet = the wrist

This matters especially with words like sin/sit/sine. Because håndled is neuter singular, the reflexive possessive would be sit:

• sit håndled

That is a natural neutral word order in Danish.

Here:

• igen modifies the action: examine again
• i morgen gives the time: tomorrow

So the sentence moves from the action to the time:

• Lægen vil undersøge hendes håndled igen i morgen

That said, Danish can move things around for emphasis, especially time expressions:

• I morgen vil lægen undersøge hendes håndled igen

Both are correct, but the original order is very normal.

Yes. Danish often fronts time expressions.

So these are both good:

• Lægen vil undersøge hendes håndled igen i morgen
• I morgen vil lægen undersøge hendes håndled igen

When i morgen comes first, Danish still keeps the finite verb in second position, so vil must come before lægen.

The sentence follows a very typical Danish main-clause pattern:

• Lægen = subject
• vil = finite verb
• undersøge = infinitive
• hendes håndled = object
• igen = adverb
• i morgen = time expression

So the pattern is roughly:

subject + finite verb + infinitive + object + adverb(s)

A key thing to notice is that Danish main clauses normally put the finite verb in position 2.

Yes. Danish forms many nouns by combining smaller words, and håndled is one of them.

It is made from:

• hånd = hand
• led = joint

So literally it is something like hand-joint, which is the wrist. This kind of compounding is extremely common in Danish.