Jeg køber en lille værtsgave, når vi bliver inviteret til middag hos nye venner.

Because Danish often uses the present tense for habitual actions or general truths.

Here, the speaker is not talking about one specific dinner invitation. They mean something like: Whenever this happens, this is what I do.

So Jeg køber means I buy / I usually buy, not necessarily I am buying right now.

Når is used for something that happens repeatedly or whenever something happens.

So:

• når = when / whenever in repeated or general situations
• hvis = if
• da = when for a specific event in the past

In this sentence, the meaning is general: whenever we are invited, I buy a small host gift. That is why når is the natural choice.

This is the passive.

• vi inviterer = we invite
• vi bliver inviteret = we are invited

The structure is:

• bliver = present tense of blive (become / get / be in passive constructions)
• inviteret = past participle of invitere

So vi bliver inviteret literally works like we get invited, but in natural English it is usually we are invited.

Yes. Vi inviteres is also passive.

But there is a difference in style:

• vi bliver inviteret sounds more common and conversational
• vi inviteres sounds a bit more formal or written

So in everyday speech, bliver + past participle is very common.

Værtsgave is a common-gender noun, so it takes en:

• en værtsgave

The adjective lille is a bit irregular. With singular nouns, Danish often uses lille:

• en lille gave
• et lille hus

So en lille værtsgave is the correct form.

Værtsgave is a compound noun:

• vært = host
• gave = gift

So a værtsgave is a host gift: a small present you bring when someone invites you to their home, such as flowers, wine, or chocolates.

A useful thing to notice is that in Danish compounds, the last part is the main noun. Since gave is the final part, the whole word behaves like gave grammatically.

Til middag is a very common idiomatic expression meaning for dinner / to dinner.

So:

• inviteret til middag = invited to dinner

If you say til en middag, it sounds more like to a dinner / to a dinner event, which is more specific and less neutral.

In this sentence, til middag is the most natural choice.

That is a very common question.

Historically, middag is connected with the middle of the day, but in modern Danish, middag often means dinner, especially in the phrase til middag.

So in this sentence, inviteret til middag is understood as invited to dinner.

Hos is used when you mean at someone's place / at someone's home / with someone in their home or care.

So:

• hos nye venner = at the home of new friends / with new friends

You would not normally use med here, because med means with in the sense of accompaniment, not at someone's place.

This makes hos the natural preposition in this sentence.

Because this is indefinite plural: new friends, not the new friends.

In Danish:

• singular indefinite: en ny ven
• plural indefinite: nye venner
• plural definite: de nye venner

So hos nye venner means at new friends' place / with new friends, not with the new friends.

Because after a subordinating word like når, Danish uses subordinate clause word order, where the subject usually comes before the finite verb:

• når vi bliver inviteret

By contrast, bliver vi inviteret? would be the order for a question.

So the sentence is following normal Danish subordinate clause structure.

Yes. If the subordinate clause comes first, the main clause follows Danish V2 word order, so the verb comes before the subject:

• Når vi bliver inviteret til middag hos nye venner, køber jeg en lille værtsgave.

Notice that it is køber jeg, not jeg køber.

That is a very important Danish word-order pattern.