Impossible badge tip: absolutely recommend following site for badge: https://gaming.stackexchange.com/questions/24904/how-do-i-infect-madagascar
I've struggled with badge one for months and, after a quick read though, managed to get it in an hour or two.
Seriously Madagascar, why the hell are you so paranoid? I mean, aside from the deadly disease that's intent on killing all of mankind.
Security guard: 'Are you a monster?'
Mardek: Rawr!
Security guard:Terrorist!!!
Turns into level fifty monster and kills entire party in one attack.
You: I THOUGH THAT WAS JUST A JOKE OPTION!!!
Badge tips: the impossible badge is essentially a skill wall plus a small amount of luck, especailly when you encounter a spring enemy which flat out sucks. The best thing to do is to just try to memorize the best path for each floor to get the most gems in the safest way, save the attract gem item for areas were there are difficult to get diamonds and don't take any unnecessary risks like trying to get a gem when there is not much tower left to move up. You can get the easy and medium badges easily just trying the impossible badge as well as most of the trophies needed for the hard badge, though you will probably need to grind a bit to finish it off and they don't register at all unless you actually exit the tower.
Impossible badge tip: try to get Natz's traits "Bunny!" and "Bunny!!!" by summoning the slime bunny a lot. The first increases its healing by 40% while the second calls it randomly like airstrike (which you should also try to get), for me I only summone it myself once in the last third of the game and the second trait kept it going until the end. It definitely made the whole the hydra boss far easier and fighting the last boss almost trivial as I didn't have to waste turns trying to spam heal and could concentrate on doing some actual damage.