Når kontrakten er underskrevet, og flyttekasserne er pakket ud, inviterer hun os op på sin altan til kaffe.

Når is the normal word for when in a future or general situation.

• når = when / once
• da = when for a single event in the past
• hvis = if

So here, Når kontrakten er underskrevet ... means When/Once the contract is signed ..., not if and not a past-time when.

These forms focus on the resulting state:

• kontrakten er underskrevet = the contract is/has been signed
• flyttekasserne er pakket ud = the moving boxes are/have been unpacked

Danish often uses være + past participle to express that something is in a completed state, or to give a passive meaning without naming who did it.

Because pakke ud is a particle verb. The particle ud often stays separate:

• at pakke ud
• pakker ud
• har pakket ud
• er pakket ud

A form like udpakket does exist, but it sounds more adjectival in many contexts. Here, the normal verbal expression is pakket ud.

Danish usually writes compound nouns as one word.

So:

• flytte
  • kasser becomes flyttekasser
• flyttekasserne = the moving boxes

English often uses two words where Danish uses one compound word.

In Danish, the definite article is usually added as an ending:

• kontrakt = a contract / contract
• kontrakten = the contract
• flyttekasser = moving boxes
• flyttekasserne = the moving boxes

But after a possessive word, Danish does not add that definite ending:

• sin altan = her own balcony
• not sin altanen

This is because Danish main clauses normally follow the verb-second rule.

The opening part Når kontrakten er underskrevet, og flyttekasserne er pakket ud takes the first position in the sentence. That means the finite verb in the main clause must come next:

• Når ..., inviterer hun os ...

If the sentence started directly with the subject, it would be:

• Hun inviterer os ...

Because sin is the reflexive possessive form. It is used when the thing belongs to the subject of the same clause.

Here:

• hun ... sin altan = she invites us up to her own balcony

If you said hendes altan, it would usually suggest someone else’s balcony, or it would sound contrastive.

Related forms are:

• sin for common gender singular
• sit for neuter singular
• sine for plural

Op på adds a sense of movement upward.

• på sin altan = on her balcony
• op på sin altan = up onto / up to her balcony

Because a balcony is thought of as a place you go up to, op på sounds natural in an invitation like this.

Til kaffe is an idiomatic way to say for coffee or over for coffee.

It refers to a social occasion, not just one physical cup. That is why there is no article. Danish often does this with meals and social activities:

• til kaffe
• til frokost
• til middag

So inviterer hun os ... til kaffe means she is inviting us over for a coffee visit.

Yes. Danish very often uses the present tense to talk about the future when the time reference is already clear.

That is true both in the når-clause and in the main clause:

• Når kontrakten er underskrevet ...
• inviterer hun os ...

In natural English, you might translate this with will invite or once the contract has been signed, but Danish does not need a separate future form here.

Yes. The sentence is understood as:

• Når kontrakten er underskrevet, og når flyttekasserne er pakket ud ...

Danish does not need to repeat når when two subordinate clauses are linked by og. The one når covers both parts.