What is the correct usage?
What can it follow? noun only?
Contrary to X, Y...
Does X and Y have to be comparable like "like/unlike"?
Thanks in advance.

Similarly, how about "whereas"?
Whereas X..., Y...
Does X and Y have to be comparable too?
Thanks in advance.

the 'X' in your writeup must be a noun, or a phrase that stands in for a noun (as in 'contrary to what they had told me, ...')

parallelism is not required. one reputable dictionary (i can't remember which, off the top of my head) lists 'contrary to orders, the soldier left camp' as an example, in which 'soldier' and 'orders' are clearly not meant to be logically parallel.

in extremely formal english usage, 'contrary to' should probably be used only as an adjective. for instance:
the financial advisor's advice, contrary to all his previous recommendations, was that his clients should liquidate their holdings in company x.
in this sentence, 'contrary to...' is an adjective phrase that modifies 'advice'.
in extremely formal usage, some experts would be uncomfortable with the soldier example above, preferring to replace it with a genuine adverbial construction such as 'in defiance of orders, ...'

'whereas' is a standard subordinating conjunction, and is used in exactly the same way as such conjunctions as 'although' or 'even though'.
parallelism is required, and X and Y must both be complete clauses.