Sometimes. In a case where the puck has been flipped into the air, a player can use their hand to either redirect it back to the ice or momentarily "Palm" the puck and move it back to the surface of the rink. Holding the puck for any longer than it takes to get it back to the ice can incur a Delay of Game Minor penalty. In the case of the Goalie, they can catch the puck in their glove or trap the puck under their gloves or pads: Both situations cause a time stoppage and a faceoff in the goalie's Defensive zone.
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You can but it is pointless because you can't hand pass in the offensive zone. You also can not "throw" it into to goal either. But you can pick it out of the air and drop it in front of you.
no, you can score with any part of your body but your leg UNLESS you do not make a kicking motion.
No, no player but the goalie can use a goalie stick. A player can knock the stick in the goalies direction that's about it. And a player can never hold more then one stick.