why does C need to consider y = 0 case and E need to consider y < 0 ?? Question has asked which of the following must be true which mean C and E are also true. Although the best fit is D, C and E are also true. Can you please elaborate a bit about this question and answer ?? I am unable to get the point.

Do you mean that all the values true for y + |y| = 0 equation must be true for option too ??
I thought of just putting the values present in options in the y + |y| = 0 equation. So I thought

For y < 0 ,
eg. when y = -2 ,
in equation y + |y| = 0,
-2 + 2 = 0
holds true.

For y = 0,
eg. when y = 0 ,
in equation y + |y| = 0,
0 + 0 = 0
holds true.

Here, if we take the condition for y = 0 then it is true for y + |y| = 0 but option C fails. And if we take the condition for y < 0 then it is true for y + |y| = 0 but option E fails.
But both conditions combined y<= 0 is true for option D as well as it is true for y + |y| = 0 .