I got the easy badge in 2010, the hard badge in 2011, and finally, FINALLY, the impossible badge in 2014! Here is a screenshot of my times, which show that you can get it even with a pretty terrible time on level e. http://prntscr.com/4a139w
Well, I finished the game, and I have to say that it was the glitchiest game I've ever played. The unresponsive jump really starts to effect whether you make it to the end in levels 3 and 4, and it becomes a game of luck of whether the jump will respond this run or not. And, to cap it all off, there was an intensely obnoxious glitch at the end. After many fails because it didn't jump, I finally made it to the fox king with all my guys and beat him, only to come to a blank screen, where I couldn't do anything and clicking the menu didn't do anything. So I had to refresh my browser and fail many times due to lag, only to beat the same final boss I'd already beaten (luckily it worked the 2nd time). This game seems like it could be good, but the lag and glitches completely ruin it.
http://prntscr.com/48qcrx
This is something like what the layout should look like going off of che1997's strategy (in best comments). If you're wondering about the flood of enemies, I just spammed the next wave button. Couldn't resist after 2000 kills :p
For the medium badge: Try to find items beginning with "Crushing" and "of Rage" because crushing adds 300 damage and of rage adds 160 (these items have a blue background). You really only need 1 of each and a weapon that can do 40 damage and equip all the items and boom, you have the badge.
The only flying object I missed was the nuke and I scored 3961467 points... I'm so glad I already had the badges, phew. (Irrelevant side note, while I was typing this comment, I looked down at my keyboard and realized there were ants all over it... Uh oh.)