Why not "is that dog alive". 'Is that an alive dog' doesnt' seem to be very good English at all!
Putting aside that the sentence in itself is weird (I think you are correct, people would say 'Is that dog alive?' in English just as they in Danish would say 'Er den hund i live?") then I think it probably boils down to this being more about finding literal translations than conveying the meaning in the other language :-)
The only situation where I could see "Er det en levende hund?" being used is if someone is presented with a display of some kind and is in doubt as to whether the apparent dog is artificial or real, and in that case most would probably rather use "Er det en rigtig hund?".
Also agree that "Is that an alive dog?" sounds completely wrong, and seemingly has been corrected since as I don't see that anywhere here now.
This isn’t a realistic sentence unless it was in a horror film- ‘Night of the Living Dog’.
If that's the case, that would be is that a real dog.