Suppose the random variable $ (X,Y) $ is independent of $ Z \equiv (Z_{1}, Z_{2}) $.
1) Is it possible that $ X $ and $Z$ are not independent?
2) Is it possible that $ X $ and $Z_{1}$ are not independent?
Since $ (X,Y) $ is independent of $ Z $, I can write \begin{align} f_{X,Y,Z}(x,y,z) = f_{X,Y}(x,y)f_{Z}(z). \end{align} I guess I can integrate $y$ from both sides of the above to get \begin{align} f_{X,Z}(x,z) = f_{X}(x)f_{Z}(z). \end{align}
The answers:
1) it is not possible
2) it is not possible
Let $F(x,y)=Pr(X<x, Y<y)$, $G(z_1,z_2)=Pr(Z_1<z_1,Z_2<z_2)$. Then for the condtional probability we have: $$ Pr(X<x, Y<y|Z_1=z_1,Z_2=z_2)=F(x,y), $$ since $(X,Y)$ is independent from $(Z_1,Z_2)$. Put $y=\infty$ and you get 1).
For another conditional probability we have: $$ Pr(X<x, Y<y|Z_1=z_1,Z_2<z_2)=F(x,y), $$ since $(X,Y)$ is independent from $(Z_1,Z_2)$. Put $y=\infty$ abd $z_2=\infty$ and you get 2).