EDIT: I was in the process, but this is what I wrote. Please add another post after this one to modify your question.
For Q1, remember that D is the electric field displacement vector, and there is no symbol for derivative anywhere in sight. The vector has components and each component is a function of x, y and z. Hence, the notation simply means evaluate the vector component at x=0 or y=0 ... etc.
For Q2, now that you understand Q1, you probably can see that those values are just a result of the substitution mentioned.
For Q3, this will also make sense once the correct substitutions are made. However, you an pretty much get to this step by just thinking in your head. You are integrating over the 6 sides of a cube. The top and bottom surfaces of the cube give zero because there is no z-component to the field (Dz=0). The Dy component does not change on the two sides that are normal to Dy, hence the vectors are equal and point in on one surface and point out on the other surface, which means they cancel out. For the two remaining sides, one is zero because the component is zero on the x=0 surface. This leaves only one surface to be calculated.