Thanks! I am not an expert in this field. What is the motivation behind considering the double integral in the first display? A priori, it seems to be quite ad hoc.

For spherical Whittaker function on $\mathrm{GL}_n(\mathbb{R})$ such asymptotics can be read off from Thereom 7.8 in Hashizume's paper "Whittaker functions on semisimple Groups."

$\pi_v$ can not be supercuspidal for all $v$, in fact, $\pi_v$ is unramified for almost all $v$. Even for $G=\mathrm{GL}(n)$ whether $\pi_v$ is tempered for almost all $v$ is a part of the Generalized Ramanujan Conjecture.

The above reference deals with the Fourier expansion with respect to the maximal unipotent (the 1+1+...+1 one). I believe the proof for the other unipotent subgroups can be similarly worked out (usually the maximal unipotent is the hardest case).

For any automorphic form $\phi$ in $\mathcal{A}^2(P)$ one can consider its constant term along any other parabolic $Q$, defined by $\int_{N_Q(F)\backslash N_Q(\mathbb{A})} \phi(n\cdot)$. For $E(g,s)$ the computation of the constant term is standard. You may look at section 3 at arxiv.org/abs/2012.07817.

You usually expect square-root cancellation, that is, the expected bound is $\ll_{q,\epsilon} B^{1/2+\epsilon}$ with polynomial dependency in $q$ which follows from some Voronoi formula. In this particular case, using the Kirillov model one can show the sum is $O(\sqrt{B})$ uniformly in $q$.

What is the support of $h$ here? Is $h$ a fixed test function? Then the sum is trivially $O(1)$. Also, you need to specify which parameters ($\pi$, $N$, $q$, $C(\chi)$, etc.) are fixed and which are varying.