Sneaking and bows really have nothing to do with each other, you exit sneak as soon as you draw. Sneak attacks are completely unrelated to the sneak damage multiplier, you get that just for hitting an unaware enemy, regardless of if you're sneaking. Many enemies you can sprint up behind and get a stab attack off before the red exclamation mark appears, granting you your 10x damage bonus.

So it's two extremes, you have to be super slow and sneak up right behind them or super fast and sprint then hit before they react.