R will affect an entire cluster of consonants of S, N, T and D. This is why the S is also /ɕ/ here.
In the place name Hornstull, for example, the middle cluster of consonants are all made retroflex by the R-sound.
"Where does the cat run?" and "Where does the cat run to?" were both accepted translations when I tried them but it's more accurate to include either "to where" or "run to" because of the t in vart. There is no rule in English against ending a sentence with a preposition. That's a Latin rule.
Yes, in formal English we would say "to where is the cat running" but ending a sentence on a preposition is now so widely accepted that many teachers and professors even allow it in writing.
So would 'var springer katten?' mean where does the cat run (ie without the to), making it location rather than direction?
Neither. It is one of the retroflex consonants in Swedish.
Here some more info: http://en.wikipedia.org/wiki/Voiceless_retroflex_stop
Note also that the sound of ch in chapter does not exist in Swedish.