Mint's minty kick mostly comes from menthol and pepper from capsaicin.
What is the equivalent dominant compound in ginger that gives it the characteristic kick?
All i can tell is it's water soluble :)
You can thank gingerol for the kick in fresh ginger, and shogaol and zingerone for the heat of dried ginger. Gingerol is chemically similar to capsaicin in chilis and piperine in black pepper, but undergoes changes when heated or dried converting it to the other compounds. You can see where gingerol and shogaol fall on the Scoville scale here.
Gingerol, named for the ginger in which it is found, are the main hot and spicy compound in ginger.
It is similar in structure to capscaicin or piperine, but is transformed in cooking into the milder zingerone.