if Y1 and Y2 are independent, does it follow that U and X are independent, where U = f1(Y1, Y2) and X= f2(Y1, Y2) If X, Y are iid rvs, and U and Z are r.v.s that can each be written in terms of X and Y, does that mean that U and Z are independent?

No. Consider i.i.d. standard normal random variables, $\,X,Y\sim\mathcal{N}\left(0,1\right)\,.$ Let $\,\,U{}={}a_1X+a_2Y\,\,$ and $\,\,Z{}={}b_1X+b_2Y$ for non-zero real numbers $a_1, a_2, b_1$ and $b_2$. 
Then, see that

$$
\mathbb{C}ov\left(U,Z\right){}={}a_1b_1+a_2b_2\ne0\,,
$$

therefore $U$ and $Z$ can't be independent.
