See also Hirzebruch, Hilbert modular surfaces. L'Ens. Math. 19, 183-281 (1973)

By Chinese remainder theorem you probably can reduce to the case where q is a prime power. Then using a twisted version of Lefschetz trace formula and Weil conjectures, the problem is reduced to bounding dimensions of some cohomology groups and in particular if you care about $q$ aspect, you should be able to get roughly square root cancellation. The keyword here is "trace functions", and you probably want to look at recent works of P. Michel, E.Kowalski et al. See for example, Kowalski's Pisa survey: people.math.ethz.ch/~kowalski/trace-functions-pisa.pdf

No, it should be ~0.641186 wolframalpha.com/input/?i=solve+(1%2F2)%5Ea+%3D+a. In particular, this is not $\zeta(2) -1 = \frac{\pi^2}{6} - 1$.

Did you look at Forster, Lectures on Riemann Surface?

Subconvexity bounds are generally on the critical line, while zero-free regions are generally equivalent to lower bounds on the edge of critical strip, so they are really different problems. As for relating bounds of L functions to bounds of various sums, these really just come from approximate functional equation and partial summation.

Isn't Dedekind zeta function always the Artin L-function of induction of trivial rep? I'm confused why you are putting them in a separate category.

It looks like you have the freedom to choose $\eta$ sufficiently small and $N$ sufficiently big. So you can certainly assume that $\mathbb{E}((v-1)1_A) = O(\eta)$ (by taking $N$ sufficiently large) with big O constant so small that when plugged into the first inequality, $\mathbb{E}(1_A) \ge 0.01 \eta^{1/2}$, by taking $\eta$ sufficiently small so that $\eta^{1/2}$ dominates. Then the result follows.

Looks like he means 2-adic local field.

Relevant: www2.math.ou.edu/~rschmidt/papers/rims2.pdf

Just to clarify, is "The issue of for which varieties the Hasse-Weil L-function equals a single automorphic L-function or when the automorphic forms involved are cuspidal is much more delicate and conjectural." the whole point of motives?