[RPG] Does deflecting missiles work against multiple missiles at once

You choose before you know the damage. Hitting and dealing damage are separate steps — if the ability was supposed to give the choice when damage was being resolved, it would say something like "… when you are damaged by a ranged weapon attack." Instead, it says "when you are hit". So when you're hit, but before damage is rolled (or revealed), you decide whether to deflect it or not.

Besides, it's a deflection. It wouldn't make sense to see how wounded you are first, and only then decide whether to make the missile hit you differently or not at all. The ability's use of a hit as the trigger instead of suffering damage as the trigger is in line with this intuition about how voluntary deflection ought to work.

No, the monk need not see the incoming missile.

...you can use your reaction to deflect or catch the missile when you are hit by a ranged attack. (PHB p.78, emphasis mine)

As you say, nothing to indicate the monk must see the incoming missile. Call it spidey-sense, or hearing it whistle through the last intervening foot, the monk can at the moment of impact choose to deflect/catch the missile.

An intentional feature of combat in D&D 5E is that you don't have facing unless you use the optional rules in the DMG. That means that "a shot to the back of the head" is more granularity than the rules cover. The six second turn in combat by necessity encompasses a variety of movement, glancing, striking, swinging, grabbing, shifting body position, and more not being described in that depth of detail ... as was done in some previous editions.