1. The position of the main stress has fluctuated over the centuries, and the OED notes that poets from Chaucer to Spenser and Shakespeare placed it on both the first and the second syllable according to need. In current English, the stress is now placed on the first syllable for the adjective and the noun, except in the meaning ‘perverse, obstinately self-willed’, in which the stress is on the second syllable, probably under the influence of the nursery rhyme beginning Mary, Mary, quite contrary.
2. The phrase on the contrary is properly used only in a statement intensifying a denial of what has just been stated or implied:

• Experience in beekeeping is not necessary —on the contrary a beginner's input can be extremely useful —Gloucester Citizen, 1999.

On the other hand denotes a differing (not necessarily opposite) point of view, and is often paired with on the one hand.
3. The phrase to the contrary is used in AmE in the meaning of on the contrary (see 2), but in BrE is used only as a mid-sentence or end-of-sentence adverbial as in There is plenty of evidence to the contrary.

Modern English usage. 2014.