Axle
Pivot
The hinge axis of something is the axle or pivot. It's the axis where things hinge.
すうじく
This is a jukugo word that uses the on'yomi readings of the kanji. You should be able to read this on your own.
「ぐるぐるバット」は、おでこをバットの先に付けて、そのバットを枢軸としてぐるぐる回転した後に、誰が目的地まで速く辿り着けるかを競う遊びです。
"Guruguru-bat" is a game where you compete on how fast you can get to the goal after spinning around a bat with your forehead on the end of the bat.