Ich schicke den Urlaubsantrag an die Personalabteilung.

Because Urlaubsantrag is the direct object of the verb schicken, and masculine nouns take den in the accusative case.

• nominative: der Urlaubsantrag
• accusative: den Urlaubsantrag

So:

• Der Urlaubsantrag ist fertig.
  Here it is the subject.
• Ich schicke den Urlaubsantrag ...
  Here it is the thing being sent.

Because an is the preposition used here for the recipient/destination of what is being sent, and in this sentence it takes the accusative case.

Personalabteilung is a feminine noun, so:

• nominative: die Personalabteilung
• accusative: die Personalabteilung

The article happens to stay die in both cases, so you do not see a change here.

Also, an is one of the German prepositions that can take either accusative or dative depending on meaning. In this sentence, it expresses direction toward a target, so accusative is used.

Because an is a two-way preposition. That means it can use:

• accusative for direction / movement toward
• dative for location / position

In Ich schicke den Urlaubsantrag an die Personalabteilung, the document is being sent to the HR department, so there is a directional idea.

Compare:

• Ich schicke den Antrag an die Personalabteilung.
  direction / destination
• Das Foto hängt an der Wand.
  location

So the accusative is used because the sentence answers the question where to?

Schicke is the 1st person singular present tense form of schicken.

Conjugation in the present tense:

• ich schicke
• du schickst
• er/sie/es schickt
• wir schicken
• ihr schickt
• sie/Sie schicken

So Ich schicke means I send or I am sending, depending on context.

Because in a normal German main clause, the finite verb usually comes in position 2.

Here the sentence structure is:

• Ich = position 1
• schicke = position 2
• den Urlaubsantrag
• an die Personalabteilung

This is the standard German V2 pattern.

If you move another element to the front, the verb still stays second:

• Den Urlaubsantrag schicke ich an die Personalabteilung.
• An die Personalabteilung schicke ich den Urlaubsantrag.

So the verb remains in the second slot, not necessarily the second word.

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

It is made from:

• Urlaub = vacation / leave
• Antrag = application / request

Together, Urlaubsantrag means a leave request or vacation request.

The -s- in the middle is a common linking element in German compounds. Learners often just need to memorize these compound forms, because the linking sound is not always predictable.

Also, in German compounds, the last part is the main noun:

• Antrag is masculine, so Urlaubsantrag is also masculine: der Urlaubsantrag

Because all nouns are capitalized in German.

So in this sentence:

• Urlaubsantrag is a noun
• Personalabteilung is a noun

That is why both begin with capital letters. This is a standard rule in German, not something special about these words.

Because Personalabteilung is feminine, and for feminine singular nouns:

• nominative: die
• accusative: die

So even though the phrase is in the accusative after an, the article still looks the same.

This can be confusing because with masculine nouns the article changes clearly:

• der Antrag → den Antrag

But with feminine nouns, nominative and accusative are identical.

Yes. Schicken is very flexible and can be used for sending:

• physical items
• letters
• documents
• files
• emails
• people, in some contexts

So using schicken with Urlaubsantrag sounds natural. In everyday office German, it can mean sending the leave request physically or electronically, depending on context.

Usually not in this sentence.

With schicken, when you mean sending something to a person, office, or department as the recipient, German commonly uses an:

• den Antrag an die Personalabteilung schicken

Zu is more common when talking about movement to a place/person in a physical sense:

• Ich gehe zur Personalabteilung.
• Kannst du bitte zum Chef gehen?

So:

• an die Personalabteilung schicken = send it to HR
• zur Personalabteilung gehen = go to HR

Yes. German word order is flexible, but the verb in a main clause still stays in second position.

Possible alternatives:

• Ich schicke den Urlaubsantrag an die Personalabteilung.
  neutral word order
• Den Urlaubsantrag schicke ich an die Personalabteilung.
  emphasizes the document
• An die Personalabteilung schicke ich den Urlaubsantrag.
  emphasizes the recipient

So the sentence can be rearranged for focus, but schicke must remain the finite verb in position 2.

It is singular here.

Die Personalabteilung means the HR department / personnel department as one department.

Its plural is:

• die Personalabteilungen

But in this sentence, it refers to one specific department.

Usually, the last part of the compound determines the gender.

So:

• der Antrag → der Urlaubsantrag
• die Abteilung → die Personalabteilung

That makes compound nouns easier to understand:

• if the final noun is masculine, the whole compound is masculine
• if the final noun is feminine, the whole compound is feminine
• if the final noun is neuter, the whole compound is neuter

This is a very useful rule for German learners.