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,
Mimics can only strictly turn into objects, that means things like, doors, door frames, key holes, keys, key rings, door knobs, knockers, peep holes- and those are just the door-and-other-doory-objects Mimics.
But...
Since a dead creature is considered an object, they can turn into a dead creature, if you want. It should be noted that they don't get the creature's stats and attacks.
Creatures that shapeshift
There are many creatures that can assume the form of humanoids and beasts without casting spells: Metallic Dragons and Deva, come to mind. I won't list them here but you can look through the Monster Manual to find the ones with the "Shape Change" trait.
A creature that can cast Shapechange is also a good candidate for what you want, it is, however, a 9th-level spell.
However, it appears what you want to be able to do is to make the mimicking creature appear stronger than it actually is. Most creatures with Shape Change can only look weaker, one exception could be the Doppelganger- taking the form of, say, an Assassin.
For Beholders, specifically...
There is a fungus called Gas Spores found in page 137 of the Monster Manual. They can serve the purpose of mimicking Beholders, as they are described as:
A gas spore is a spherical, balloon-like fungus that resembles a beholder from a distance, though its true nature becomes increasingly obvious as one approaches it.
Have fun freaking out your party!
Best Answer
RAW, weapons stick to the mimic
The ability you quote does say that mimics stick to "anything" that touches them. Moreover, the description of the mimic specifically states (emphasis added, MM220),
Therefore, RAW, any weapon that hits the mimic will stick to it. That will make fighting it with a melee weapon pretty annoying, but PCs generally have access to attacks that aren't affected by this ability, such as spells and ranged attacks. Also, it's only sticky as long as it's mimicking something, and loses its stickiness in its amorphous form.