What are some deep results in sampling theory?
As you  are looking for some deep results, let me mention only two:


the Kadec's-$1/4$ theorem that gives a stability bound under which  perturbed exponential Riesz bases remain Riesz bases (M. I. Kadec, The exact value of the Paley-Wiener constant, Dokl. Akad. Nauk SSSR, 1964, see for instance  R. Young, An introduction to non-harmonic Fourier series, p. 42) 
the theory called "finite rate of innovation" (Sampling Signals With Finite Rate of Innovation, IEEE Trans. Signal Proc., 2012, M. Vetterli et al.) which essentially generalizes Nyquist-Shannon theory to non band-limited signals, requiring "twice the number of degrees-of-freedom" per unit interval, like the two-points per period in the classical cases, for some parametric signals (finite number of spikes, piece-wise polynomials, etc.). 


I like the genericity and the simple bounds for both examples. 

You can probably  find more ideas in Classical and approximate sampling theorems; studies in the $L^p(R)$ source and the uniform norm, J. Approx. Theor., 2005, P. L. Butzer et al. and Sampling Theory, a Renaissance: Compressive Sensing and Other Developments, 2015, G. E. Pfander editor. In the latter, you might like Chapter 9 - Sampling in Euclidean and non-Euclidean Domains: A Unified Approach (sphere, Hyperbolic Space), and Chapter 10 - A Sheaf-Theoretic Perspective on Sampling, where you get some topology.

There are all sorts of exiting developments in zero- or level-crossing, single-bit sampling, which I do not know enough to be more precise: round-off, jitter, truncation, saturation and aliasing errors are real problems.