These are all wrong, unfortuntely. You can't use what you are trying to prove in your proof.
Everyone has said basically, let x + -(-x) = 0 but that is exactly what you are trying to prove!
Here's the proof (using only field axioms):
0 = 0
-(-x) + [-(x) + x] = [(-(-x) + (-x))] + x
-(-x) + 0 =...