På fødselsdagskortet lykønsker vi hende med det nye job, og bagefter pakker vi gaven ind i gavepapir.

This is because Danish is a V2 language in main clauses: the finite verb normally comes in second position.

• Vi lykønsker hende ... = normal order, with the subject first.
• På fødselsdagskortet lykønsker vi hende ... = a different element is moved to the front, so the verb stays second and the subject comes after it.

So after a fronted phrase like På fødselsdagskortet, Danish requires:

• På fødselsdagskortet
  • lykønsker
    • vi

not:

• På fødselsdagskortet vi lykønsker ...

Here på means on in the sense of something written on a card.

Danish often uses på for writing or information appearing on a surface or medium:

• på kortet = on the card
• på listen = on the list
• på skiltet = on the sign

Using i would suggest something more like inside the card physically, which is not the main idea here.

Danish makes compound nouns by joining words together, so:

• fødselsdag = birthday
• kort = card
• fødselsdagskort = birthday card

The ending -et is the definite ending for a neuter noun, so:

• et fødselsdagskort = a birthday card
• fødselsdagskortet = the birthday card

So the whole word means the birthday card.

This is a very common Danish pattern.

When a noun is definite by itself, Danish often adds the definite meaning as an ending:

• kortet = the card

But when there is an adjective before the noun, Danish usually uses a separate definite article:

• det nye job = the new job

So:

• fødselsdagskortet = the birthday card
• det nye job = the new job

You do not normally say:

• det nye jobbet

The verb pattern is:

• at lykønske nogen med noget

That means:

• to congratulate someone on something

So in this sentence:

• lykønsker vi hende = we congratulate her
• med det nye job = on the new job

Even though med often means with, after lykønske it matches English on in meaning.

Because hende is the object form of the pronoun.

• hun = she
• hende = her

Here she is receiving the action of the verb lykønsker, so Danish uses the object form:

• Vi lykønsker hende = We congratulate her

not:

• Vi lykønsker hun

This is about adjective forms.

The noun job is a neuter noun:

• et job = a job

In the indefinite singular, a neuter noun takes an adjective in -t:

• et nyt job = a new job

But in the definite form, the adjective usually takes -e:

• det nye job = the new job

So:

• et nyt job = a new job
• det nye job = the new job

That shift from nyt to nye is normal.

Bagefter means afterwards / after that.

Just like in the first clause, Danish uses V2 word order in the second clause too. When bagefter is placed first for emphasis or sequencing, the verb must come second:

• Bagefter pakker vi gaven ind
• not: Bagefter vi pakker gaven ind

After og, Danish starts a new coordinated main clause, so the V2 rule still applies.

You could also say:

• ... og vi pakker bagefter gaven ind — grammatically possible in parts, but much less natural here
• ... og bagefter pakker vi gaven ind — natural and clear

Because pakke ind is a verb + particle combination, similar to English wrap up.

In a main clause, the finite verb appears earlier, while the particle often stays later in the clause:

• Vi pakker gaven ind = We wrap the gift up / We wrap the gift

So the parts separate:

• pakker = finite verb
• ind = particle

This is very common in Danish with verbs like:

• tage ... med
• skrive ... ned
• stille ... op

There is also a single-word verb indpakke, but pakke ... ind is very common in everyday Danish.

Here gavepapir means wrapping paper, and it is being treated like a material/substance noun, much like paper in English.

So Danish naturally says:

• pakke gaven ind i gavepapir = wrap the gift in wrapping paper

No article is needed because the point is the material, not one specific sheet.

If you wanted to be more specific, you could say things like:

• i noget gavepapir = in some wrapping paper
• i det gavepapir, vi købte = in the wrapping paper we bought

Because the sentence contains two main clauses:

1. På fødselsdagskortet lykønsker vi hende med det nye job
2. bagefter pakker vi gaven ind i gavepapir

In standard Danish punctuation, a comma is usually written between coordinated main clauses, even when they are linked by og.

So the comma helps mark the boundary between the two full clauses.