This is not location. The cat is not currently where you are asking about, he's going TO that location. You are asking in which direction the cat is going to get to that location.
No. In questions, the verb needs to come before the subject. A simple question would be like this: Springer katten? - 'Does the cat run?' But if there's a question word, like where, what, who etc, you just put that before the verb.
no, that would translate to 'Where to the cat runs' which is just strange. Such construction might be seen in poems or similar. You need to put the verb 'springer' as the second word.