Random Walk expectation and independence
Solution 1:
Hint for 1: If $x_n=n$, then you necessarily have $y_n=0$. More generally, information about $x_n$ gives some information about $y_n$.
Hint for 2: note that $E[|P_n|^2] = E[x_n^2] + E[y_n^2] + E[z_n^2]$ so it suffices to compute each term. In turn, $x_n$ is the sum of i.i.d. random variables that take values $1, 0, -1$ with probabilities $1/6, 2/3, 1/6$ respectively, so you can expand $x_n^2$ as the square of a sum of these random variables and proceed.