Well, my fingers are going to hurt after this...

For simplicity's sake, I'll write the current name of the achievement and then the new suggested name and description; writing out all of the names and descriptions would likely break the character entry limit. If an achievement isn't listed, then I didn't think the current state was bad enough to demand a rewrite.

Also, you might want to consider adding "in X mission" for all the mission-specific Wanted Men or tasks. Just replace the X with whichever mission it would be.

"Put X Out Of Action" sounds weird; these should be swapped to "Defeat X, in Y mission".

Lastly, spoiler warning for anyone else reading through this. Many of the achievements reveal which characters join your team or give hints about future conflicts.

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Abandoned children
Abandoned Children
Completed the first episode!

A Person Who Doubt
A Person Full Of Doubt
Defeat Sanson, a manager in the Drifter Clowns.

Aren't you...mommy?
Are you... my mommy?
Tame a Draki.

Auto Refill
Exhaust all of Kylie's protocols, including Overcharge, while using her Engineer class.
(Mirror this on her Hacker class achievement, as well, but replace "Engineer" with "Hacker".)

Burned out Mountain of Fire
Dormant Volcano
Defeat the Volcano Crabmit.

Chakra Openning
Chakra Opening
Exhaust all uses of every equipped Ninja ability with Misty in a single mission.
(Spelling error fix.)

Change of a friend
Evolve a beast.

Citizen Interests First
Rescue all civilians in 'Overlapping coincidence, another phrase for "destiny"' on Hard or higher.

Cleaning Up The Mess
Hide a defeated enemy in the scenario mission 'Shadow Of The Past'.

Cold-blooded Sword
Defeat the wanted man Ryung-Lee.

Dark Hunter
Defeat Jeff, the hunter.

Don't underestimate me!
Stun an enemy using one of Bianca's abilities.

Endlessly falling rain
Endless Rain
(Most of the "Complete X mission" achievement names match the mission name but lack capitalization.)

Eureka
Eureka!
Unlock a new hidden mastery through mastery research.

Everything as I planned!
Equip a new ability.

Fate that nearly missed
Almost forgotten by Fate

Find some gold!
Activate 'Exchange of Equivalents' 100 times.
(Use numbers whenever possible. Also, this was my least favorite achievement to get, by far.}

Four-leaf Clover
Activate the 'Luck' status 10 times.

Gather up~
Heal at least 3 units simultaneously with Anne's Starfall.

Greedy
Equip 2 mastery sets on 1 character.

Headshot!
Defeat an enemy with a headshot from Giselle.

Hero's Moves!
The Ferocious Red Panda
Use at least 4 chained abilities with Irene's Overcharge skill.
(I propose swapping the titles of these achievements. Chaining 4 abilities is much more ferocious than only chaining 2.)

Hide Yourself Good
Hide Yourself Well
(Grammar fix. You verb well, but a noun can be good. Do well, be good.)

Hit 'em all once again!
Defeat at least 3 enemies simultaneously with Irene's Flame Explosion.

Hitman
Defeat the wanted man Shandian.

I need gas.
Use all of Ray's support sprays in a single mission.

Impossible Challenge
Defeat Marco in the 'Pugo Shopping Street' mission.

Indestructible Relationship
Reach the repair squad with Heixing first in the 'To Make Matters Worse' mission.

Into The Cumulus
Dodge a ranged attack while in a smoke grenade's cloud.

Jet Black Destroyer
Defeat the Schwarz Destron.

Karma
Defeat an enemy with Leton's counterattack in (whichever mission he is first playable in).

Leave It To Me
Leave it to me!
Reach an 'Exalted' bond with any character.

Let's twist!
Defeat Oliver, the dancing wanted man.

Line up!
Defeat at least 3 enemies simultaneously with Albus' Gale Slash.

Line up once again!
Defeat 3 or more enemies simultaneously with Sion's Flash Beam Cannon.

Mass Production
Craft 100 items.

Molten Mountain of Ice
Melted Snowcap
Defeat the Snow Mountain Crabmit.
("Molten" is not really a state for a frozen mountain, but snowcap mountains do sometimes melt.)

My Friend
My Hunting Buddy
Complete 10 battles with both Giselle and a summoned beast active.

My Friend
Forge a bond between any characters.

No more boring performance
Get off the stage!
Use Finale with no Coolness stacks.

Nonstop Cheers
With at least 1 Coolness stack, use Finale with at least 8 filled Performance Slots.

One Kick, One Kill
Defeat an undamaged enemy with Leton's Frost Storm Kick.

One Shot, One Kill
Defeat an undamaged enemy in a single hit.

Out of Fuel
Oil Crisis
Make an enemy machine run out of fuel.

Pink Panther
Defeat Pink Horn.

Pure White Destroyer
Defeat the Vice/Alabaster (whichever name you end up using) Destron.

Rain of Shine
Take a bow!
(This will probably make a bit more sense to English speakers.)

Researcher
Create 100 masteries via researching.

Return safely to the base
Made it back, safe and sound.
Complete 10 missions without any company members being defeated.

Reverse Engineering
Successfully use Kylie's Hack skill 3 times.

Safety First
In the 'To Make Matters Worse' mission, escape with Kylie and don't lose any allies on Hard difficulty or higher.

Shouting of Angry Bull
Raging Bull
Defeat Scott in 'A Spreading Magenta'.
(Grammar fix and a length snip for a really long achievement title.)

Smuggler
Defeat the notorious smuggler, Brad.

Snatching
Steal one of Jeff's beasts with Giselle's Tame ability.

Sniper in darkness
A shot from the dark.
Cause the Confusion state on an enemy with Giselle's Shadow Sniping.

Sorry I'm Late!
Lose at least 1 civilian in any mission. Oops.

Staying out of trouble!
Catch them all!
Successfully tame 10 beasts with Giselle.

Tactical Decision
Stun at least 3 enemies simultaneously with a Wind Pressure Grenade.

The Agent
Defeat Tyrone, the swindler.

The Broker
Defeat the shameless broker, Cheche.

The Cleaning Agent
Defeat the deadly "cleaner", Felix.

The Duty of the Leaved Men
The duty of those left behind.

The Ferocious Red Panda!
Hero's Moves!
Activate an ability chain with Irene's ultimate move.

The Formula Worker
Extract 100 Psi-Stones.

The King of Mud
Defeat the fierce Spotted Negoori.

The King of the Mist
Slay the King Draki.

The Leaved Man
(I don't actually know what to change this to, but it sounds odd in English.)
Defeat the Drifter Clown officer, Hanson.

The legend of heroes
The Heroic Legend
Defeat 100 hero rank enemies.

The Legendary Warrior
Defeat 100 legend rank enemies.

The Queen Of The Culvert
Defeat the Ashen Yasha Monarch.
(I propose shortening the name to "Ashen" since it takes up less space and gets the idea of their appearance across. Also, most large spiders in reality are female; males tend to be tiny and short-lived.)

The Madding Sword
The Maddening Sword
Defeat the lethal swordsman, Yuranxiang.

The Man Who Stepped Back
Bring justice to Roberto with Irene in the 'To Not Regret' mission.

The Operator
Defeat the shadowy operator, Ivan.

The Queen of Mud
Defeat the Albino Negoori queen.

The Queen of the Mist
Slay the Draki Queen.

The Shadow Of Truth
Successfully use Bianca's Illusion ability 5 times.

The Turncoat
Defeat the burning wanted man, Hellfire.

This is what I created.
Build a robot with Kylie.

This path is my path.
Change a character's class.

Trance
Maintain the Trance status for 10 turns.

Trap Architect
Set 10 traps with Giselle.

Treasure Hunter
Open 100 loot chests during missions.

Twinkling Moment
Defeat the sparkling wanted man, Twinkle Jack.

Vestige of World War
Forgotten War Machine
Defeat the modified war robot, Red Eye.

Walpurgis Night
Defeat at least 3 enemies simultaneously with Anne's Pouring Starlight Arrows. She'll feel guilty about it.

Watcha Doin'!
Defeat at least 3 enemies simultaneously with Heixing's Annihilation Shot. True Bullet Hell!

We can take care of anyone for you.
Defeat the fearsome duelist, Ryo Ryuzaki.

We deliver anything.
Defeat both Aizia and Garry in the same mission.

Why this code is here?
Why is this code here?
Use Kylie's protocol abilities 50 times. Beep boop.

Why'd you come?
Why did you even try?
Defeat an enemy with Alisa's Forestallment mastery.

Wild Life
Ignore 100 field effects with Wild Life.

You can nothing
You can't defend yourself.
Use Alisa's Neutralize Counters mastery to ignore a counterattack.

You guys ever heard about deep sea before?
(Another case where I actually can not figure out what the intent is but I know it sounds weird in English.)
Reach 300 Action Time. You'll be waiting for awhile!

You did the thing!
You successfully read this entire list! Congratulations!