$V'$ is indeed dense in $V$. Taking Fourier transforms, note that any bounded measurable function with compact support is the Fourier transform of a function in $V$. And the Fourier transform of $f(x)-f(x-c)$ is $(1-e^{c\xi})\hat f(\xi)$. For the clincher, note that the space of bounded measurable functions with compact support contained in the complement of $(2\pi/n)\mathbb{Z}$ is dense in $L^2$. (Adjust signs and factors $2\pi$ accroding to your taste in Fourier transform conventions.)

$V'$ is indeed dense in $L^2$. Taking Fourier transforms, note that any bounded measurable function with compact support is the Fourier transform of a function in $V$. And the Fourier transform of $f(x)-f(x-c)$ is $(1-e^{c\xi})\hat f(\xi)$. For the clincher, note that the space of bounded measurable functions with compact support contained in the complement of $(2\pi/n)\mathbb{Z}$ is dense in $L^2$. (Adjust signs and factors $2\pi$ accroding to your taste in Fourier transform conventions.)