If X, Y independent then X/Y and Y are independent? if two random Variables $X, Y$ are independent 
Does that mean that $\left(\frac{\ X }{Y}\right)$  and $Y$ are independent? 
for example is it true that E[$\left(\frac{\ X }{Y}\right)$|$Y$]= E[$\left(\frac{\ X}{Y}\right)$] because of the independence of the random variables X and Y?

Suppose that $X,Y$, satisfy $\mathbb{P}( X=-1     ) =\mathbb{P}( X=1     )=1/2 $ and
$\mathbb{P}( Y=-1     ) =\mathbb{P}( Y=2     )=1/2 $.
Set $Z=X/Y$, then $\mathbb{P}( Z=1/2, Y=-1)=0$ however 
$\mathbb{P}(Z=1/2)\neq 0$ and $\mathbb{P}(  Y=-1)\neq 0$.
So, they are not independent. 
For the conditional what we have is $E\left ( X/Y\right |Y=y ) = E(X/y)$
