Suppose I have a pseudo-differential operator on $\mathbb R$ whose symbol is
$a(x,\xi) = e^{-\xi^2/2}$
(notice, no dependence on $x$)
Question 1: Are the following statements correct?
$a$ is an Hörmander symbol in the class $\mathcal S^{m}_{1,0},\; \forall\, m\in \mathbb R$. Since $\mathcal S^{m}_{1,0} \subset \mathcal S^{n}_{1,0}$ for $m\leq n$ holds, then we might say $a\in \mathcal S^{-\infty}_{1,0}$ where we define $\mathcal S^{-\infty}_{1,0}=\cap_{m\in \mathbb R}\mathcal S^{m}_{1,0}$.
Question 2: Does $a$ have a principal symbol? If it does, what is it?
N.B. For the notation one might refer to the Wikipedia page on pseudo-differential operators.
The answer to both of your questions is yes, since the function is Schwartz. The rapid decay of Schwartz symbols makes the first statement immediately true. For the second, consider the definition of the principal symbol as given in Taylor (you mentioned it in your first edit). For a Schwartz function, you can check that you can take every term in the asymptotic expansion to be $0$ (note that $a$ is always in the appropriate class, as you noted in part (a)). In particular, the principal symbol is zero.