Danach checke ich das Aufgabegepäck ein und bekomme die Bordkarte.

Because German main clauses follow the verb-second rule.

In this sentence, Danach is placed first, so the finite verb checke must come second, and the subject ich comes after it:

• Danach checke ich das Aufgabegepäck ein.

If you start with the subject instead, you get:

• Ich checke danach das Aufgabegepäck ein.

That is also grammatical, but the emphasis is a little different. Starting with Danach highlights the sequence: after that.

Because einchecken is a separable verb.

In a main clause, the prefix ein- separates from the verb and goes to the end:

• Ich checke das Gepäck ein.

But in the infinitive, it stays together:

• Ich möchte das Gepäck einchecken.

And in a subordinate clause, it also stays together:

• ..., weil ich das Gepäck einchecke.

So checke ... ein is just the normal split form of einchecken.

Yes. It is a loanword from English, but it is very common in German, especially in travel contexts.

You will hear things like:

• einchecken
• online einchecken
• am Flughafen einchecken

That said, for baggage, many speakers also use more traditional German wording such as:

• Gepäck aufgeben = to check in luggage
• Ich gebe mein Gepäck auf.

So the sentence is understandable, but depending on context, a more idiomatic version might be:

• Danach gebe ich mein Gepäck auf und bekomme die Bordkarte.

Aufgabegepäck means checked baggage or luggage that you hand over at the counter.

It is understandable, but in everyday speech many people would simply say:

• Gepäck
• mein Gepäck
• aufzugebendes Gepäck in more formal contexts

So:

• das Aufgabegepäck

is not wrong, but it can sound a bit more formal or technical than:

• mein Gepäck

A very natural sentence would be:

• Danach checke ich mein Gepäck ein und bekomme die Bordkarte.

Because the gender of a German compound noun is determined by its last part.

Here, the last part is Gepäck, and Gepäck is neuter:

• das Gepäck

So the whole compound is also neuter:

• das Aufgabegepäck

This is a very useful rule for German compounds.

Because Karte is a feminine noun:

• die Karte

So die Bordkarte is also feminine, because the last part of the compound is Karte.

Also, in this sentence it is the direct object of bekomme:

• ich bekomme die Bordkarte

In English, you might naturally say my boarding pass, but German often uses the definite article where English prefers a possessive:

• ich bekomme die Bordkarte
• ich wasche mir die Hände

If you want, meine Bordkarte is also possible, but die Bordkarte sounds perfectly normal.

Both are in the accusative case, because they are the direct objects of the verbs.

• checke → what do I check in?
  das Aufgabegepäck
• bekomme → what do I receive?
  die Bordkarte

In this sentence, the forms happen to look the same as nominative:

• das stays das in nominative and accusative neuter
• die stays die in nominative and accusative feminine

So the case is accusative, even though the article does not visibly change here.

Because German often uses the present tense to describe:

• routines
• planned actions
• instructions
• a sequence of events

So:

• Danach checke ich das Aufgabegepäck ein und bekomme die Bordkarte.

can mean something like:

• After that, I check in the baggage and get the boarding pass.

This works very much like English when describing a process:

• First I go to the counter, then I check in my bag, then I get my boarding pass.

Both can often be translated as then or after that, but there is a small difference.

• dann = then
• danach = after that / after it

Danach points more clearly to the previous step. It sounds a bit more explicitly sequential.

So both are possible:

• Danach checke ich das Gepäck ein.
• Dann checke ich das Gepäck ein.

In this sentence, danach slightly emphasizes after that step.

It is understandable, but many German speakers would choose slightly more natural phrasing.

Very natural alternatives include:

• Danach checke ich mein Gepäck ein und bekomme die Bordkarte.
• Danach gebe ich mein Gepäck auf und bekomme die Bordkarte.

Why?

• mein Gepäck sounds more everyday than das Aufgabegepäck
• Gepäck aufgeben is a very common airport expression

So the original sentence is not wrong, but it sounds a bit more textbook-like or formal than the versions above.

Because all German nouns are capitalized.

So in this sentence:

• Danach is capitalized because it is the first word in the sentence
• Aufgabegepäck is capitalized because it is a noun
• Bordkarte is capitalized because it is a noun

This is one of the most noticeable spelling differences between German and English.