WIth a rock. Thrown at high speed, of course. Or, just shoot it. Or, just throw a really heavy log on it. Or, just cut its head off. Or, pick it up and swing it around really fast and bash its head off of a tree.
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Yes. Copperheads are venomous snakes. A copperhead's venom is milder than other pit vipers but it's enough to kill a small animal like a puppy. They regularly kill and eat rats. The bite may or may not kill the pup but will likely make it sick and perhaps lose a leg.
You could, but you shouldn't try to, as you risk being bitten a second time. Instead you should seek immediate medical attention.
Why would you want to kill it? Copperheads are beneficial and should only be killed if they pose an immediate threat.