1. On the contrary is not especially spoken English. It introduces a statement that says the opposite of the last one. “He did a bad job, his boss should have been very unhappy; on the contrary he smiled and told him he would do better next time”. And yes, you can say for example “he worked hard and was promoted, on the contrary she was lazy and got the lowest salary”. Here's a good link to see many examples on how to use it : http://www.wordnik.com/words/on%20the%20contrary/examples 2. I just discovered that website, and it's actually quite useful. Here's also the definition and how to use after all : http://www.wordnik.com/words/after%20all. I don't think it's unusual, I use it often, and I've heard many people use it too. 3. Regarding “but” and “to be fair”, it's not exactly the same. “To be fair” = to be honest, or you can use it as “to his defense” (We lost the game because of him but to be fair it was the first time he played. I could just have said 'but', but that wouldn't have had exactly the same meaning, you wouldn't have this “to his defense” idea in it). But look at the website, reading examples is always the best way to understand something ;) 4. Can't think of a good answer to last question right now (actually I have an idea, but I'd rather not say anything than say something that would mislead you, let's just wait for a native to explain it well). I didn't explain everything because I think the website does it well, and I don't have much time right now, but if it is not clear enough, I'll explain it more later.

1. "On the contrary" is normally used to say someone is wrong, usually in an argument or discussion: Bob says "There is no evidence for global warming!" Anne says "On the contrary, there is plenty of evidence." But it would probably only be used in a fairly formal academic discussion. Down the pub, people would instead say "Rubbish! There is plenty of evidence!"