Wiener's tauberian theorems seem relevant to your results (in fact this has been discussed recently on mathoverflow here: Is there an L^p tauberian theorem?Is there an L^p tauberian theorem?). One of these theorems states that the linear combinations of the translates of a function, f, is dense in L^2(R) if and only if the Fourier transform of that function doesn't vanish on a set of positive measure. Since the Fourier transform of the Gaussian is a Gaussian (and doesn't vanish) this is very similar to your Theorem 1. Of course, Wiener's theorem only gives you an approximation of the form $ \sum_{n=1}^{N} a_{n} e( (x- t_n)^2)$ for some sequence of real numbers $t_{n}$. You might be able to recover your result by approximating the the sequence $t_{n}$ by a sequence of the for m $\{nt\}$ (after possibly adjusting the coefficients a_n), however this isn't immediate to me.
If these remarks sound along the lines of what you are looking for, Wiener's book "the Fourier integral and certain of its applications" is a good reference. Jacob Korevaar's Tauberian Theory: A Century of Developments has a much more comprehensive (and current) treatment of the subject.
