"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.
if you're really wanting literary language, "Whither runs the cat?" is fine, if perhaps a bit formal/dated.
One does not tend to use whither / hither / thither (or whence / hence / thence) in spoken English these days outside set phrases like "I was running hither and thither" or as the compound "henceforth".
I think I'd just drop the "to" at the end of your example, "Where is the cat running" is most natural : "to" is understood.