I don't know what you mean by "decided to use the R2" The domain is not a "choice"! The problem says "f(x, y, z)" so f depends upon the three variables x, y, and z. The domain of the function is a subset of $\displaystyle R^3$. Of course to be able to take that square root we must have $\displaystyle z- x^2- y^2\ge 0$. What is that set?
The "level sets" of levels 0 and 2 are the graphs of $\displaystyle \sqrt{z- x^2- y^2}= 0$ and $\displaystyle \sqrt{z- x^2- y^2}= 2$ which are equivalent to $\displaystyle z= x^2+ y^2$ and $\displaystyle z= x^2+ y^2+ 2$. What are those graphs?