I have three independent but non-identically distributed random variables $X_1, X_2, X_3$. I define two new random variables for the comparisons of $X_1$ against $X_2$ and $X_3$, call them $P = (X_1 > X_2)$ and $Q = (X_1 > X_3)$. Are $P$ and $Q$ necessarily independent because the $X_i$s are independent?
No, $P$ and $Q$ are not necessarily independent. A single example suffices:
Let $X_1$ be $0$ with probability $\frac34$ and $100$ with probability $\frac14$.
Let $X_2$ be uniformly distributed on $(10,20)$ and let $X_2$ be uniformly distributed on $(40,46)$.
Then $P$ and $Q$ are completely correlated: If $P$ is true, so is $Q$.