Relationship between z-transform and DFT
For a) you're correct. For b), $x_1$ is a length $2N$ signal, and its DFT is given by

$$X_1[k]=\sum_{n=0}^{2N-1}x_1[n]e^{-j2\pi kn/2N}=\sum_{n=0}^{2N-1}x_1[n]e^{-j\pi kn/N}\tag{1}$$

With $x_1=x[n]+x[n-N]$ you get

$$\begin{align}X_1[k]&amp;=\sum_{n=0}^{N-1}x[n]e^{-j\pi kn/N}+\sum_{n=N}^{2N-1}x[n-N]e^{-j\pi kn/N}\\&amp;=X\left(e^{j\pi k/N}\right)+\sum_{n=0}^{N-1}x[n]e^{-jk\pi(n+N)/N}\\&amp;=X\left(e^{j\pi k/N}\right)+e^{-jk\pi}X\left(e^{j\pi k/N}\right)\\&amp;=X\left(e^{j\pi k/N}\right)(1+(-1)^k)=\begin{cases}2X\left(e^{j\pi k/N}\right),&amp;k \text{ even}\\
0&amp;k\text{ odd}\end{cases}\tag{2}\end{align}$$

Eq. $(2)$ shows that decimation in the frequency domain corresponds to aliasing in the time domain.