Trots startavgiften vill vi teckna mobilabonnemanget.

Because Swedish follows the V2 rule in main clauses: the finite verb usually comes in the second position.

Here, Trots startavgiften is placed first as an adverbial phrase. Once that takes the first position, the finite verb vill must come next, and the subject vi comes after it.

So:

• Vi vill teckna mobilabonnemanget. = normal order
• Trots startavgiften vill vi teckna mobilabonnemanget. = the fronted phrase takes first position, so the verb moves before the subject

This is very common in Swedish.

Trots means despite or in spite of.

It is followed by a noun phrase, not a full clause. In this sentence:

• trots startavgiften = despite the start-up fee

Compare:

• Trots regnet gick vi ut. = Despite the rain, we went out.
• Även om det regnade gick vi ut. = Even though it was raining, we went out.

So:

• trots
  • noun phrase
• även om
  • clause

Startavgiften is the definite form, meaning the start-up fee.

Swedish often uses the definite form when talking about a specific thing that is understood from the situation. Here, it refers to the particular fee attached to that mobile plan.

Compare:

• en startavgift = a start-up fee
• startavgiften = the start-up fee

So the sentence is talking about a known, specific fee, not just any fee in general.

Mobilabonnemanget is the definite form: the mobile subscription / the mobile plan.

This usually means a particular subscription is being discussed, one that both speaker and listener know about.

Compare:

• ett mobilabonnemang = a mobile subscription
• mobilabonnemanget = the mobile subscription

So the sentence means they want to sign up for that specific plan, not just any mobile plan.

Because Swedish makes compound nouns very freely, much more often than English does.

Here:

• mobil = mobile
• abonnemang = subscription
• mobilabonnemang = mobile subscription / mobile plan

Then the definite ending is added:

• mobilabonnemanget = the mobile subscription

In English, we usually write these as separate words, but in Swedish they are often written as a single word.

Good question. Teckna has several meanings depending on context.

In everyday Swedish, teckna can mean:

• draw
• sign
• take out or sign up for an agreement, policy, or subscription

In this sentence, teckna mobilabonnemanget means:

• sign up for the mobile plan
• take out the mobile subscription

This use is common with things like:

• teckna ett abonnemang = take out a subscription
• teckna en försäkring = take out insurance
• teckna ett avtal = enter into a contract

So this is a fixed, idiomatic business/service verb.

Not usually, at least not if you want the most natural phrasing.

Köpa means buy, but a mobile subscription is normally something you sign up for or take out, not something you simply buy. That is why Swedish commonly uses teckna.

So:

• teckna ett mobilabonnemang = the natural, idiomatic expression
• köpa ett mobilabonnemang = understandable, but less precise and less natural in many contexts

You might use köpa for a phone itself:

• köpa en mobiltelefon = buy a mobile phone

But for the service contract, teckna is better.

It is a noun made up of:

• start = start
• avgift = fee, charge

Together:

• startavgift = start-up fee / initial fee

Then the definite ending -en is added:

• startavgiften = the start-up fee

The head word is avgift, which is a common gender noun (en avgift), so the definite singular form is avgiften.

In a main clause, inte usually comes after the finite verb and after the subject if the verb has moved into second position.

So:

• Trots startavgiften vill vi inte teckna mobilabonnemanget.

Here the order is:

• Trots startavgiften = first position
• vill = finite verb in second position
• vi = subject
• inte = negation
• teckna mobilabonnemanget = infinitive phrase

This placement is very typical in Swedish.

Yes, that is possible, but it sounds a bit different in emphasis.

• Trots startavgiften vill vi teckna mobilabonnemanget.
  This puts strong focus on despite the start-up fee.

• Vi vill trots startavgiften teckna mobilabonnemanget.
  This is also grammatical, but the emphasis is a little less front-loaded.

Swedish often moves elements to the front to highlight them. So the original sentence sounds very natural if the speaker wants to stress the contrast.

No. Even though they look similar, vill usually means want (to), not future will.

So:

• vi vill teckna = we want to sign up / we want to take out

If you want to talk about the future in Swedish, you often use the present tense or sometimes ska rather than a direct equivalent of English will.

So be careful: vill is about desire, not just future time.

Because it comes after vill, which is a modal verb.

After modal verbs like:

• vill = want to
• kan = can
• ska = will / shall / going to
• måste = must

the next verb is normally in the infinitive form, without att.

So:

• vi vill teckna
• not vi vill att teckna

This is similar to English want to sign up, except Swedish does not use a separate word corresponding to to in this structure.