I can confirm that the game is always 100% solvable, and it's always possible to get a clear 100% screen BUT you do need some luck, especially in the early parts. you NEED to find the extra experience point chests sooner rather than later, you need to get some lucky shots in the early/random phase, and you need some luck to get the nome as early as possible BUT after you get up to 8 or 9 hearts it's purely a matter of skill to kill the dragon. I can't count how many times I messed up near the end because I'm getting tired and "brain no do math good no mo".
After I got good I used the cheat code to explore every question and secret, and still with total understanding of every aspect i STILL love playing to pass the time, truly a great game.
Also if you want the mouse stamp you need some SERIOUS luck not to accidentally hit a rat before you clear some space, or SUPER lucky to hit the rat king with the first few hits. The other 2 passive stamps are far easier to skill it out.