1) Is it true that if X, Y are independent and X, Z are also independent then X, Y + Z are also independent?
2) If W, Y and W, Z are independent (and if the previous point hold, namely that W, Y+Z are independent), is it true that also X + W, Y + Z are independent?
1) is false. In the following link there is an example of events $A,B,C$ such that the events are pairwise independent but not jointly independent: Example of Pairwise Independent but not Jointly Independent Random Variables?
Let $X=I_A, Y=I_B$ and $Z=I_C$. Then the hypothesis is satisfied but $X$ is not independent of $Y+Z$ for the following reason: $P(X=1,Y+Z=2)=P(X=1,Y=1,Z=1)=P(A\cap B\cap C)$ and $P(X=1) P(Y+Z=2)=P(X=1) P(Y=1,Z=1)=P(A)P(B)P(C)$. Since $P(A\cap B\cap C) \neq P(A)P(B)P(C)$ it follows that $X$ is not independent of $Y+Z$ .
