Can you cheat by purposefully missing the opponent’s conker? Thus reducing the total impact on conkers in this match vs conkers in other matches and getting an advantage?
You take it in turns to hit, moving yours on their turn is cheating. Given that you have a lot more control over what kind of hit happens on your turn than on theirs, skipping your turn is never going to give you an advantage (unless your opponent is somehow anti-competent).
I can't imagine so. A match is over when one persons conker is destroyed. If you were purposefully missing, you'd be throwing the game. Theirs would still take a battering from hitting yours as well.