An der Paketstation scanne ich den Code, hole das Paket ab und stelle mein Handgepäck kurz neben mich.

An der Paketstation means at the parcel locker / parcel station.

The preposition an is a two-way preposition, so it can take either:

• dative for a location: at, by
• accusative for movement toward something

Here it describes a location, not movement, so German uses the dative:

• die Paketstation → dative: der Paketstation

So:

• an der Paketstation = at the parcel station

If it were about moving to it, you could get accusative instead:

• Ich gehe an die Paketstation. = I go to the parcel station.

Because German puts the finite verb in second position in a main clause.

The sentence begins with An der Paketstation, which takes the first position. That means the verb must come next:

• An der Paketstation | scanne | ich ...

So the normal ich scanne becomes scanne ich after the opening phrase.

Compare:

• Ich scanne den Code.
• An der Paketstation scanne ich den Code.

This is very common in German. When something other than the subject comes first, the verb still stays in second position.

Because Code is the direct object of scannen, so it takes the accusative case.

The noun is masculine:

• nominative: der Code
• accusative: den Code

So:

• der Code = the code (as subject)
• den Code = the code (as object)

In this sentence, I scan the code, so the code is what gets scanned, which makes it the direct object.

Abholen is a separable verb, and here it means to pick up or to collect.

Its two parts are:

• holen
• prefix ab-

In a main clause, separable prefixes move to the end:

• Ich hole das Paket ab.

So in your sentence:

• hole das Paket ab = pick up the package

This is different from plain holen, which often just means get/fetch.
Abholen often has the idea of collecting something from a place.

Because all three verbs share the same subject: ich.

The sentence is listing actions done by the same person:

• scanne ich den Code
• hole das Paket ab
• stelle mein Handgepäck ... neben mich

In German, once the subject is clear, you usually do not repeat it in a coordinated list unless you want extra emphasis or clarity.

So this is natural:

• ..., scanne ich den Code, hole das Paket ab und stelle mein Handgepäck ...

It would also be grammatical to repeat ich, but it would sound more repetitive:

• ..., scanne ich den Code, ich hole das Paket ab und ich stelle ...

Mein Handgepäck is in the accusative because it is the direct object of stellen.

You are placing the hand luggage somewhere, so the hand luggage is the thing being moved.

Handgepäck is a neuter noun:

• nominative: mein Handgepäck
• accusative: mein Handgepäck

So the form does not change here, which is why nominative and accusative look the same.

This is a very common German question.

Neben is a two-way preposition, so it can take:

• dative for location: next to
• accusative for movement toward a position: to next to

Here the verb is stellen, which implies movement from one place to another. You are placing the hand luggage beside yourself, so German uses the accusative:

• neben mich = to a position next to me

Compare:

• Das Handgepäck steht neben mir.
  The luggage is standing next to me.
  → location → dative
• Ich stelle das Handgepäck neben mich.
  I place the luggage next to me.
  → movement/change of position → accusative

Here kurz means something like briefly, for a moment, or for a short time.

So:

• stelle mein Handgepäck kurz neben mich
  means
• I place my hand luggage next to me for a moment

It does not mean short in the physical sense here. It is an adverb describing the duration of the action.

German often distinguishes placement verbs more specifically than English.

• stellen = to put/place something in an upright position, or more generally to set something somewhere
• legen = to lay something down, usually horizontally
• setzen = to set someone/something into a sitting position

With luggage, stellen is very natural if you are setting a bag or suitcase down beside you, especially if it is upright.

So:

• Ich stelle mein Handgepäck neben mich. = very natural

If the luggage were being laid flat, legen could also be possible in another context.

Yes. A Paketstation is usually an automated parcel locker or parcel pickup station where you can send or collect packages.

For a learner, it helps to think of it as a compound noun:

• Paket = package/parcel
• Station = station

German forms many nouns this way by combining words.

The commas separate items in a series of coordinated actions:

• scanne ich den Code,
• hole das Paket ab
• und stelle mein Handgepäck kurz neben mich

This works like listing actions in English:

• I scan the code, pick up the package, and place my hand luggage next to me briefly.

In German, commas are commonly used between coordinated main-clause parts in this kind of list, especially when the parts are longer.

Yes, that is grammatical, but it changes the emphasis slightly.

Compare:

• An der Paketstation scanne ich den Code ...
  Emphasis first on where the action happens
• Ich scanne an der Paketstation den Code ...
  Emphasis first on I

German is flexible with word order, but the element placed first usually gets more focus or serves as the topic of the sentence. Starting with An der Paketstation sounds very natural if the location is the frame for everything that follows.