Here is the full list of all 14 Open Me! (JP) trophies - there are 11 bronze, 2 silver and 1 gold.
Earn a S rank with every single box
"You" are the last reward
Earn a "Checker Flag" with "One Box Car"
Earn a bronze trophy with "Puzzle Box"
Earn a bronze trophy with "Snap Snap Snap Box"
Earn a bronze trophy with "Super Hinge Box"
Earn a bronze trophy with "Rebel Box"
Earn a bronze trophy with "2 Count Box"
Earn a bronze trophy with "Thunder Box"
Earn a bronze trophy with "Multi-purpose Surveillance Box"
Earn a bronze trophy with "Puzzle Box?"
Earn a bronze trophy with "Master Box"
Earn a bronze trophy with "Clock Box"
Earn a bronze trophy with "Windmill Box"