I'm not sure if I understand your question Sparrow.
I think what you want to know, is that if your rider dies, will your mount still attack? The answer to that is yes, but they will only attack if they are still around other friendly units, once they are left to themselves they will become tame-able so you may capture them.
Not sure how shooting/riding will pan out, as it may be hard to do it interface-wise. Because if you just click on a target, your archer should just shoot, not run and shoot. I suppose you could have a little option (with a hotkey) and you could setup a range for them to run, kind of like a guard patrol in Warcraft.