Fidel Dungeon Rescue

This can be unlocked as early as the first level. If you approach a chest and back (CTRL/LT) it will flash open and you'll be able to see it's contents. If it's a mimic (has teeth) rather than being filled with gold approaching it from the rear will defeat it and unlock the achievement.

This achievement can be attempted in Level 4 but as the levels are randomly generated even if you get a level with four plants the achievement may still not be possible. Before you can eat a plant you need to eat all of the flowers/vines surrounding it. As such any plants that have four vines may leave you trapped. You can use bombs to clear the vines and any other enemies but need to eat the plants (bombs will clear the tiles above, below, left and right of the squares they are used on but leave the diagonal tiles untouched). The achievement will unlock upon eating the fourth plant so you don't need to leave yourself an exit.

In Level 10 (where the Dragon Claw snatches Grandma) you need to bark at all five candles to blow them out and create a portal to the secret level and then enter the portal. Upon entering the level you will unlock the Secret Level achievement. You'll unlock the Checkmate achievement upon taking the last pawn as shown in the level below. Then simply retrace your steps to a point where you can exit the level.

I've included an image to show my route but I'm not sure if this level has a fixed solution or if it's random like the others.

This is achieved on the alien rave level and requires you to clear all of the small aliens. The achievement will unlock upon clearing the last one so you don't need to leave yourself an exit. You can also use bombs if necessary. I recommend trying to level up either before entering this level or early on in the level to give you more choice in your movements and when you need to use health packs/potions. My solution is shown below as an example but the level is randomly generated.