1) Can a monk with a two handed weapon still deflect arrows when not on his turn, by using free actions to free one hand at the end of his turn and reapplying his hand to the two-handed weapon at the beginning of it?
It would presumably mean that AoOs would have to be with unarmed strikes.
2) Am I right that you need to have Shield Proficiency to use a quickdraw shield, so a monk that wants to use such a shield when it isn't his turn will need two feats: shield proficiency and quickdraw?
On the second one, because you put your shield away at the end of your initiative/turn, and then you're no longer wielding it so you *do* get your AC whilst everybody else is busy trying to hit you.
A monk loses his WIS to AC and the AC bonuses from class if he wields a shield. Thus losing your Monk AC to gain the shield AC. Often this will be a bad trade and result in a loss of AC overall.