Well, I learned linear algebra from a book by Halmos (also measure theory from another): I recovered from one book when I learned tensor products properly, and the Steinitz lemma; probably not from the other one! It is too hard to give solid recommendations. But now you can look up definitions online, it is important to have books with concepts and "loose ends" to research, not just definition-lemma-theorem-proof. Books with suggestive content.

The behaviour of a prime q depends only on the order of the cyclic subgroup it generates in that (multiplicative) group. The generators of the whole group mark out certain residue classes, and so the q are those that lie in certain arithmetic progressions.

I suppose the Sieve is basically of interest (now) in the context of an interval [N, N + M] in which one wants to find all primes, given an initial set of primes. You are framing it as the (supposedly original?) iterative procedure for finding the next prime from a seed set. There is not much point arguing over formulations, though. This sieve is easy to apply for practical machine computation, in some cases; and generally hard to apply in theoretical studies as it stands. Hard to see the benefit of your remark. Every is free to philosophise about primes, naturally ...

A way of thinking is in terms of equations a = b, or to be clearer F(x) = F(y). If you give me as axioms such equalities for all pairs of elements of your connected sets, the question for me becomes the scope of what can be proved about F. So my description is: use the partition induced by the values of F being provably equal. Your remark is to do with the sets of axioms. So this is a kind of series of remarks about "equational logic", in my view. The gluing on overlaps is accurately modelled by the conjunction of F(x) = F(y) and F(y) = F(z), which gives F(x) = F(z).

en.wikipedia.org/wiki/Tsen%27s_theorem and references. The Brauer group is indeed trivial. I suppose this is proved in geometric analogues of class field theory.

Indeed, if you want general background, reading the research literature would be a last resort. The sheer number of papers has a great deal to do with library budgets, also. The remark that growth is exponential does support the idea that the change might come rather suddenly. (By the way Cartier once said something like "Serre has no idea what a Laplacian is", which is not to be taken literally but an indication of non-universality in a very versatile algebraist.)

I think Theory of Algebraic Invariants, Hilbert lectures recently translated and reprinted for CUP, shows why we might still take Hilbert seriously as a geometer, given that he clearly had a sophisticate's view of moduli problems.

Well, I answered the question as I understood it, with the word "most" of mathematics interpreted as it would have seemed to a mathematician in the year X. In those days all the important guys could meet at the ICM. I think it is incorrect to "read back" our understanding to the 1910s, and I came to this view by reading a solid history of mathematics through, hitting Harald Bohr as a "pain barrier". Algebra wasn't the major issue right then: arguing that in our terms they should have felt it was risks anachronism. Of the Hilbert problems only 14 and 17 are close to our abstract algebra.