The Center has more responsibilities offensively and defensively traditionally speaking. He would support the puck in the offensive and defensive zone in front and behind the net as well as support the wingers on their respective side. Traditionally the center men were the more creative, talented player.
Wingers job were to skate up and down their respective wing and you are correct about what is traditionally a right or left-winger. You could receive passes easier if your sticks curve was inward to the passed puck but you would have less shooting angles on the goalie. If you were good at receiving the puck pass on your backhand you would be coming in on the goalie with the puck in the center of the ice instead of the side, therefore more angles to score.
This is all the traditional theory of hockey though. Players are so much more skilled and the trap system has changed the traditional roles somewhat, but that's the jist.
Checking can be done whether you have one, two or no hands on the stick. You can’t take more than 3 strides before checking someone or else that’s charging. You can’t check from behind, you can’t raise your arm (much) as you check that becomes elbowing.
To check up on the checking rules, go to NHL.com and look under rules… I’m sure they have a link to rules and whatnot there.
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