The Blue Slime is tough, no doubt. For a Soldier, you should just keep plugging away with upper body attacks (hopefully you have the Dirty Paintbrush, but it should be possible with just a Small Feather). Yeah she can heal herself in many ways, but the Soldier should have an easier time than some squishier classes.
For the abandoned shack, the picture gives you the clues you need, as well as the sign.
"V for Victory, The Powerpuff Girls won a Gold Medal"
V for Victory is a hang gesture that involves 2 fingers.
The Powerpuff Girls are a superhero team with 3 members.
A gold medal is the 1st place reward.
So you need to pull switch 2, 3, and then switch 1 (in that order. Every time you hear a really high laugh, that means the pattern is reset. You have to work in groups of three.)
The picture shows you which three switches are the right ones AND tells you what number they are.
The Star points at every switch except the correct 3.
The trapezoid points at all the wrong switches.
The triangle points at the three correct switches.
Blossom is the 1st Powerpuff Girl (Pink arrow points to Switch 1)
Bubbles is the 2nd Powerpuff Girl (Blue arrow points to Switch 2)
Buttercup is the 3rd Powerpuff Girl (Green Arrow points to Switch 3)
So pull Switch 2, then Switch 3, then Switch 1.