1. Это о чём-то говорит. Это что-то значит.
2. I think there is no single equivalent. There are some examples:
to come off victorious - выйти победителем = "to exit as a winner"
to come off with honour - выйти с честью = "to exit with honour"
to come off cheap - дёшево отделаться = "to shake off cheap"
to come off clear - выйти сухим из воды = "to exit dry out of water"
to come off best in a fight - победить соперника в бою = "to win one's rival in a fight"
to come off a loser - остаться/оказаться в проигрыше = "to be left with a loss, to find oneself with a loss"

For this phrase "I met her parents finally, but I made a lot of dirty jokes and came off stupid" I would suggest "я выглядел глупо" (I looked silly).
And for the first one, "he always came off worse in an argument" - does it mean he always gives up easily? or that he is always won in an argument? or, is it saying "he always turns mad in an argument"?