"What is your cat doing on the table?"
Translation:Что твоя кошка делает на столе?
"it is just relatively stiff, so we did not think of this particular word order"
I couldn't miss the humour in this: As someone coming from English learning Russian, I tend to think in English sentence structure by default. As a result I've had my fingers burnt very often when I structure a Russian sentence from an English perspective only to discover the Russian version is actually front to back.
Now in this particular case the English and Russian versions are in lock step and those of us that have learnt to muck around the words to get the Russian version right ended up with the "stiff sounding" version. ))
The simplistic answer is that Duolingo uses a grammar-translation method and you are not going to learn how Russians put sentences together from that. I recommend taking the Английский course. The further along you are in the Russian course, the easier the Английский course will be. I highly recommend it. It's an "eye opener".
It is arguable that the object of на in this sentence could be accusative, since there is activity involved?
From various articles on prepositions and case, I've collected three rules regarding на:
1. на [prepositional]: "on, in, at" to define location
2. на [accusative]: "onto, on" when movement is involved
Say, the cat is eating a mouse on the table:
кошка ест мышь на стол(е)?
On-line translators say it's still prepositional, and that you need something like "she puts dishes onto the table" in order to create the need for на стол. That's not the same as something happening while things are already on the table.
2a. на [accusative] when на expresses "for' + [circumstance/activity], e.g., "for breakfast" = на завтрак:
We don't know it's feminine from English, but the default cat in Russian is feminine. (Not just in Russian; I used to have three male cats and one female, and I had a friend who kept referring to the boys as "she", because his default cat was apparently also female.) If you want to say that it's a tomcat, you can use кот.
As confirmed by DemitriusV above, you can change word order to indicate emphasis. Here's how I would write it: Что твоя кошка делает на столе? = What is YOUR CAT doing on the table? Что на столе делает твоя кошка? = What is your cat doing ON THE TABLE? Что делает твоя кошка на столе? = What is your cat DOING on the table?