I'm not sure why you want to prove this by contradiction - your proof is exactly the same as the direct proof, so it adds nothing in terms of understanding. In general, it is inadvisable to use proof by contradiction if a direct proof is just as easily obtained. This is because if you make any mistakes in your working, passing that mistake off as a contradiction gives an invalid proof, even though the conclusion is correct.
For the uniqueness of additive inverse proof, there is a lot of things that may not be well defined. What is (-v)? Is it a third additive inverse? Is it w? Is it u? If (-v) is just an arbitrary additive inverse, does the symbol (-v) represent the same element on both sides? Without knowing uniqueness (which is what you want to prove), how can you add (-v) on each side of the equality without showing they are the same?