Learn German: Bring mir die Leine und die Bürste! - Bring me the leash and the brush!

This is an informal command (the imperative) given to one person, like a friend or family member. To tell someone to do something, you start with the du form (bringst) and drop the -st at the end, leaving just Bring.

Why do we use mir instead of mich here?

mir is the dative form of ich (I). When you bring someone something, the person receiving the item is the indirect object, which takes the dative case. The things being brought (die Leine, die Bürste) are the direct objects, which take the accusative case.

If I wanted to reply by asking 'Which leash?', how would I use welch- here?

You would ask Welche Leine? Because Leine is a feminine noun and acts as the direct object (accusative) in the context of the action ('Which leash [should I bring]?'), the word welch- takes the feminine accusative ending, which is -e.

How can I tell what case die Leine and die Bürste are in, since die doesn't change?