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.
A grappled creature can use its action to escape.
It says an action, so it requires an action. If it could be done as one of several attacks, it would say so, just like it says for making a grapple.
If you're the DM, you can houserule this, of course, but think carefully before doing so, because allowing one attack of Extra Attack to escape grapples vastly favours the players. There are some monsters that are only dangerous because of grappling, so making it easier to escape a grapple is fairly crippling for them. Also, due to the way Multiattack works, this will only make it easier for players to escape monster grapples, not for monsters to escape player grapples.
Best Answer
yes, they can use either Strength (Athletics) or Dexterity (Acrobatics)
PHB, p 195:
Just as in their attempt to fight the grapple with either Athletics or Acrobatics, they can choose either in their Escape. The grappler must use their Athletics.