As I said, I'm not sure about "putting someone in the wrong" - I don't think it's used much around here. It may be old or regional.
As for "do me wrong" and "do me right" - both are informal, so you wouldn't usually use them in a report or speech. Tazzler is right; it does mean to treat someone badly (do me wrong) or well (do me right), but as I said before, it's subjective. Some people will say "He did me wrong!" over a minor issue, but usually it's used for major things.
If you use the "in context" link from Panjandrum's link for "do me wrong" (at the top of the page under the links for Spanish, French, and Italian) you'll see the second link is of a man whose dog bit off his toe because "I think he had a sense that he knew (the infection) was going to do me wrong..."
In this case, the infection would have made him very sick and might have killed him.
In the song I mentioned earlier, the singer means the man who left her and broke her heart.
For "do me right," again, it's usually used for more significant things. If an insurance agent sends you the paperwork for your recently-burned house, that's just expected. If you've been accused of burning it yourself and he helps prove that you're innocent, you could say he really did you right.
"Do me right" is much less common than "do me wrong." A more formal version would be "He wronged me," but that's also not used very often. Note that "He righted me" is NOT used in this sense at all.
More often we would be specific about what someone did to us:
"He cheated me."
"She robbed me."
"He hit me."
"She set me up."