Here is the full list of all 14 Slimesphere achievements.
Win with all allies alive
Kill an enemy with fall damage
Force an enemy into the spikes
Deal 15+ damage at once
Win with all allies untouched
One-Hit kill with a ranged unit.
Cause 2+ falls in a row
Recover 20+ HP in a single battle
Kill 5+ enemies during a round
Win a battle with a team of Rangers
Cause 3+ falls in a row
Win a battle with only one ally unit
Block a Ranger from attacking
Hit 3+ enemies in a row w/ a Mage