The negative of 0 in any given vector space is (−1)0.

This statement is true. Can anyone explain why. Is it because the negative of anything is determined by the inverse axiom, so the negative of y is (-1)y and the negative of -1 is (-1)-1?

The reason is that scalar multiplication is involved in finding the negative vector - and it does not matter whether it is 1 *0 or -1 *0 , the answer is still 0: the zero vector is its own negative.