if I have $f(x,y,z)=\cdots $ and two constrictions $g(x,y,z)$ , $h(x,y,z)$
and I need to calculate the min/max etc.
(1) I know that I can do a Lagrange system with $5$ equations and $5$ variables.
(2) if I do two systems separately (one with $g(x,y,z)$ and the other one with $h(x,y,z)$) will it give me the same results as if I were to do them as point 1?
(3) if my $f(x,y,z)=z$ and let $g(x,y,z)$ be the constraint. Can I still do Lagrange multiplier even though $f_x=0$ and $f_y=0$ ?
(2) Of course not (in general). Suppose that $f(x,y,z)=x^2+y^2+z^2$, that the first constrain is $x=-1$ and that the second constrain is $y=-1$. If you solve two systems separately, then the solution of the first system will be $(-1,0,0)$, whereas the solution of the second one will be $(0,-1,0)$. None of these points belong to the line $x=y=-1$. The solution will be $(-1,-1,0)$, of course.
(3) Yes, you can.