When asking this question, in principle one should make a distinction between two cases:
- When $n=1$, the space $PW_{S}$ is a space of entire functions of exponential type, with all of their unique properties.
- The case $n\geq2$ is much more involved and as far as I know is not fully understood, some results were obtained by Olevskii and Ulanovskii ("On multi-dimensional sampling and interpolation").
A proof to the fact that a uniformly discrete set forms a Bessel sequence can be found in thm 17, on chapter 2 in Young's book "An Introduction to Non-Harmonic Fourier Series" (the case $n=1$).
The fact that a sampling set is always relatively dense follows from Landau's necessary condition (appears in "Necessary density conditions for sampling and interpolation of certain entire functions"), and applies to both cases.