If I have one card left in my deck and my opponent exhausts a character for Fall of Oa's effect, can I activae Birthing Chamber in response to draw my last card and keep Fall of Oa's effect from happening?

If I do this, does my opponent still get to put his deck into his hand?


Another question..

If I already have no deck, can my opponent use Fall of Oa's effect anyways, and still put his deck into his hand?

you will get to draw the card, but the way Fall of Oa is worded, your opponent will still get to put his deck in his hand