Express the surface using two distinct parameterization methods where $x^2+y^2=z^2$ for $z\in [0,1]$
Ans: I can parametrize it as $$\vec r(t,z)=(z\cos(t), z \sin(t), z)$$ for $z\in [0,1]$ and $t\in [0, 2\pi]$.
Another parameterization may be as follows $$ \vec r(u,v)=(u,v,\sqrt{u^2+v^2})$$ but how to define the ranges of $u$ and $v$? Is there any other possible parameterization?
You are rewriting your surface as the graph of a function. Your surface in question is a cone. If you look it from above/below, you see a disk of radius $1$, without holes: this means that the domain of your graph (i.e. the range of $(u,v)$) is $(u,v) \in \{ (x,y): x^2+y^2\leq 1\}$.
Having a visual understanding of your surface helps.