Translation:The king and the queen live in the castle.
Your link no longer works but here is another from wikipedia for those interested: https://upload.wikimedia.org/wikipedia/commons/0/0f/Drottningholm_kollage_2011abc.jpg
The picture is from Drottningholm Castle, just outside of Stockholm. The actual Royal Palace in the centre of Stockholm is no longer used as a place of residence, but for representation and such affairs of state, and as a museum. And for the Royal family to stand and wave to the masses on National Day, June 6th, of course. :p
If I remember correctly from my visit, to the right would be the royal theater, one of the oldest theaters still operating (according to my tour guide!). The royal family lives in the left wing of the castle and you can tell if they're home based on which flag is flying . Princess Victoria lives in Hagaparken in Haga slott though! :)
Prepositions are extremely difficult and don't always make sense. When it comes to residence it depends on where it is. Usually smaller places take the preposition "i" whereas larger and more public places take the preposition "på". This is not a rule though, just a guideline. I'll give you a few examples:
Han bor på slottet - He lives in the castle
Hon bor i huset - She lives in the house
Han bor på hotellet - He lives at the hotel
Hon bor i stugan - She lives in the cottage
So your sentence above would have to use "i";
Presidenten bor i (det) Vita Huset
Remember that you have to use "vita" in the definite form.
I may have asked this before but I don't know that I got an answer. In a sentence like this it would be fairly usual in English to drop the second "the" in the sentence even when we mean the definite of both.
Eg The knife and fork. The king and queen. The cat and dog. etc.
In this sentence does drottning have to be in the definite or could it follow a similar rule to English where either makes sense and both imply the definite?
Since Swedish uses a suffix for determined, it wont work ("kungen och drottning bor..." is wrong).
When using den/det, it changes meaning.
"Den röda och snabba bilen" is one car that is red and fast.
"Den röda och den snabba bilen" is two cars.
That said, it can happen that people say "Den röda och gröna bilen" to mean two cars, even if it's not correct.
I HATE the translations to write in English. I KNOW that it is THE king and THE Queen but the second "the" is redundant in English. If I write The King and queen live in the castle that is exactly the same meaning, we just dont use the redundant second "the". Its not going to be The king and a queen because she is queen by virtue of marriage to the king whom she is living with. And its clear its "the" queen without saying the "the" because you are talking about the place BOTH live which is why we dont say it! This is really annoying when you say write in English and I write in how I would say it in English and then because its not an exact word for word translation its wrong but the English is right! this is so frustrating at times when the translations have no bearing on English as its spoken by native speakers.
When I answered this, I did choose "The king and THE queen" from the translation options although the repetition of the word "the" makes the translation a bit more swenglish/svengelska. I do think in cases like this (in a language course) it's important to translate things literally from the Swedish as a reminder that the swedish word "drottning" should take the definive article (drottningen).