[RPG] Who wins a Grapple contest if the checks tie

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.

Although contests are RAW, how to run a contests is explicitly made the DM's responsibility via their own good judgment of the specific situation at hand. That means that, RAW, as soon as a contest is on the table (so to speak), it's in the DM's hands to make any and all adjustments necessary for it to make sense.

In this way, the rules are — again, by RAW — not telling the DM how to do it, only giving them a basic tool to riff off as necessary. Some situations will call for using that judgement to give a stunned character the chance to oppose a grapple, while most situations will probably not.

It might have been nice to have corner cases like this nailed down instead of punted to the DM, but relying on DM judgement is intended to be a feature of 5e, so this is quite intentional. DMs are smarter about the situation in front of them than a rulebook could ever be, and 5e's design intent is to harness that constructively.