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, and there's a trivial counterexample (or example, depending on how you interpret it): $U=Z=X+Y$ (or any other expression, even a constant, instead of $X$ and $Y$).
