[RPG] Should a monk be allowed to use acrobatics to “shove”

No, your Dexterity bonus can't be used for grapple checks instead of Strength.

You don't get the swapping benefit for everything you do unarmed — being unarmed is just the first condition of the ability. You get the effect, when unarmed, for "unarmed strikes and monk weapons."

Starting a grapple isn't striking (aka hitting to deal damage), it's grabbing them to hold on.

The basic principle is that not all melee attacks in the game involve actually hitting someone in the "striking" sense. There are lots of melee attack rolls called for in the rules that are there to resolve something other than a strike.

Now, if you want to throw someone in a martial-arts style after a successful grapple, or grab someone to throw them instead of to impose the Grappled Condition, that's an entirely different ball of wax because grappling doesn't offer that option, but there is another rule that does.

For that you actually want to look at the improvisation rules (PHB, p. 193), which invoke the DM's Contest rules (DMG, p. 238), in order to resolve a throw. If your throw technique involves speed and agility to maximise leverage to use the opponent's own mass and strength against them instead of using your own Strength, that would reasonably qualify as a Dexterity Check for your half of the Contest.

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.