> You are assuming that additive inverse is unique

Suppose that some number a has two additive inverses,x and y.

    a + x = 0 = x + a

    a + y = 0 = y + a

Consider

    x + a + y = (x + a) + y = 0 + y = y
    x + a + y = x + (a + y) = x + 0 = x

Thus x=y, and so the additive inverse is unique.

  > you are assuming that -1(-1) is some value at all;

We can define it to have a value and then derive the properties of the value. We can show it to be consistent by creating an explicit model. Such things are relatively easy to do, but require a level of detail inappropriate for this context.

  > you could write an extremely similar proof that
  > leaves you with 0/0 = 1 and it would be faulty to
  > conclude that 0/0 actually is 1.

Could you present such a proof for us?