"It appears to the naïve (me) that there is no nontrivial restriction on the primes for which we know there remain an infinite number in the restricted sequence. Q1. Is this superficial perception in fact true?"
No, it is definitely not true. Here are a few examples:
(1) Let $a>0, b$ be two relatively prime integers. Are there infinitely many prime of the form $an+b$? Yes.
(2) Let $P(X)$ be a monic polynomial of degree $n$ with coefficients in $\mathbb{Z}$. Are there infinitely many prime $p$ such that $P(x)$ has $n$ distinct roots mod $p$? Yes.
(3) Let $X$ be a projective smooth variety over $\mathbb{Q}$, $\chi$ the Euler-Poincaré characteristic of the manifold $X(\mathbb{C})$, $n$ an integer. Are there infinitely many prime $p$ such that the the number of points of $X(\mathbb{F}_p)$ is $\chi$ modulo $n$? Yes. Same question with $X(\mathbb{C})$ replaced by $X(\mathbb{R})$? Yes.
(4) Are there infinitely many primes $p$ that can be written $a^2+b^2$? Yes. $a^2+8b^2$ with $b$ odd? Yes...
(5) Let $a,b$ be two integers (such that $4a^3+27b^2 \neq 0$), $a_p$ the number of solutions of $y^2=x^3+ax+b$ modulo $p$, minus $p$. Are there infinitely many primes $p$ such that $a_p=0$? yes. Are there infinitely many primes $p$ such that $a_p \neq 0$? Yes. Let $\alpha$ and $\beta$ be two reals between $-1$ and $1$, and $\alpha$ smaller than $\beta$; are there infinitely many primes $p$ such that $\alpha < a_p/2 \sqrt{p} < \beta$? Yes.
One could multiply those examples. They all belong to algebraic number theory, and a line of thought that has begun with Dirichlet's theorem (example 1), and has developed into the modern theory of algebraic number fields, Galois representations, automorphic forms and the Langlands program. Perhaps the most salient result is Cebotarev's density theorem, of which (1) is a very special case, (2) is a consequence, (3) also a consequence in combination with Grothendieck's étale cohomology, (4) also a consequence. Only (5) lies really beyond this result, due respectively to Noam Elkies, Jean-Pierre Serre, and the long list of people responsible for the proof of Sato-Tate.
Admittedly, there are many natural and interesting sequences of integers in which we can reasonably conjecture that there are infinitely many primes, and to which this line of thought is not supposed to apply (Mersenne's primes, to name one).