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, it's not.

When lifting the other character you are either taking an action, or you interacting with an object -- depending on your DM's rulings.

If the DM rules that the pile driver is an attack (which I'd argue it is), that would cost you an action, and you can only repeat it if you have movement and an action to spend on it. This assumes you've grappled them, and have an attack left.

If the DM rules you are use an object. If it wasn't an enemy, you'd only be allowed to pick it up once. Once you drop it (which is free), it is no longer a free action to pick it up again. Use An Object action:

You normally interact with an object while doing something else, such as when you draw a sword as part of an attack. When an object requires your action for its use, you take the Use an Object action. This action is also useful when you want to interact with more than one object on your turn. (PHB 193)

Since you can't lift someone and do something else, it might take a full action to lift someone above your head. Remember that a round is 6 seconds, dropping someone more than once (or twice if you have the actions for it) would take about that long. Moreover, really can't lift someone and jump while doing something else, so it could be argued it can't be an interaction and must be a "Use Object" action.

To rule otherwise would allow a free 5d6 damage without using any action, and that, while awesome the first time (and a DM may allow it once, by Rule of Cool), is game breaking.