Assuming z=-2 and x=-1
why assuming? why do you need to make an assumption here?
in general picking numbers is a bad way to prove sufficiency (it is an excellent way to show insufficiency). maybe it works for -1 and -2.... but what about -20,-24? and -100,-102? would you go to check all possible answers? i guess not....
in the moderate-difficult gmat questions, it would happen that the most straightforward "number picks" would work... but there will be "corner cases" which wouldn't, and are more difficult to find.
i wouldn't suggest using picking numbers to show sufficiency. it is logically incorrect - and will inevitably lead you to make mistakes.
here is how i'd show sufficiency of st2:
sumithra starts right by inferring from stem, that if y is positive then
z<x<0 and hence |x-z|=x-z
also: x<0 so |x|=-x, z<0 so |z|=-z
according to this it is always true that:
|x-z|+|x| = x-z-x = -z = |z|
simple algebra that proved the sufficiency of st.2