1. On the contrary is fine - someone says something you disagree with, you can say "on the contrary, I think he DID do it". In contrary - No 2. On the conclusion - No. In the conclusion is possible - in the conclusion (of the play) it became clear who had committed the murder. "In the conclusion' has the sense of some duration. 'At the conclusion', is probably used more frequently to indicate a point in time. At the conclusion of the performance the dancers received much applause. If you are giving a presentation/speech, you can say "In conclusion, I would like to say....". this indicates to the audience that you are finished your speech and leaving them with a final idea/summary.

With both sentence sets, only the first phrases are correct. 1. On the contrary, the Chinese trade surplus has increased since the trade wars begun. 2. In conclusion, the trade wars are destructive to both countries' economies. (Do not use "the" in "in conclusion.")