A: Yes, you can use the power of the Batarang before your draw. In a way, that’s kind of the point. Of course you have to have already played it on a previous turn, since you can’t play it to the table before you take your main draw. It is possible, however, to draw for your turn, play it, use it’s power, and then do something which causes you to increase your draw, or draw extra cards.
For example, you could use the Egads! Rule (draw a random card off the top of the deck and play it immediately) at any time during your turn, so you might gain that top card in that way as well – of course you’d have to use it immediately, so make sure you want to do that.