Since you are just finding a volume and not integrating a function over a volume the integral will be
with the limits added
You can do the integration in any order you like, I think the easiest is z then y then x
As you rightly pointed out, the limits on the z integral are 0 to 4+x2
The limits on the y integral are got from the equations y= 4-x2 and x+y=2 which will be y= 4-x2 and y=2-x. Now to decide which is the lower and which is the upper limit, the lower limit will have the smaller value. 4-x2 is always less than or equal to 2-x so 2-x should be the upper limit.
The limits on the x integral are also got from the equations y= 4-x2 and y=2-x. Putting them equal to each other you find that 4-x2=2-x The solutions to that equations are the limits for x.