Here is a basic question that confuses me: why can’t we reduce factoring to D-Wave problem?

We certainly can. But if we do, then there’s no reason whatsoever to expect any speedup (and even D-Wave might agree about that).

Factoring is in NP, and is believed to be “NP-intermediate.” In particular, NP-complete problems can’t be reduced to factoring unless NP=coNP, but factoring can certainly be reduced to any NP-complete problem (including D-Wave’s QUBO problem).

The issue is that no one has ever found reducing factoring to a general NP-complete problem, and then trying to solve the latter, to be a competitive way to factor numbers. When you do so, you “throw away” all the number-theoretic structure that makes factoring easier than many other search problems! For example, using the Number Field Sieve, it’s possible to factor an n-bit integer classically in ~2^{O(n^1/3)} steps. But how is your SAT-solver supposed to know about that? Likewise, even if the D-Wave machine were capable of running Shor’s algorithm (which it isn’t), if you first reduce factoring to QUBO and then run the adiabatic algorithm on the latter, the algorithm isn’t going to “magically recognize” that the QUBO instance came from factoring and that it can therefore do something much faster. Instead, it will be stuck doing an exponential brute-force search, and I would be astonished if it were anywhere near competitive with clever classical algorithms like the Number Field Sieve.

On your last question, I’d say that we still don’t understand what the D-Wave machine does, well enough to be able to talk about what complexity class might correspond to it. The D-Wave machine implements stoquastic Hamiltonians only, and those are believed to not give us all of BQP. If the temperature is too high, you might not even get anything outside of BPP. But the devices are a moving target: as soon as you figure out the limitations of one machine, D-Wave announces that maybe nothing you say is relevant to their next machine. So for the time being, the relevant question is just whether you can get any empirical speedup over classical computers in a fair comparison—not what complexity class their machine corresponds to.

Scott stipulates  “For me personally, the central question is whether or not I can at least see a “straight path forward” to getting an asymptotic [what does this word mean?] speedup over any possible classical algorithm, under mathematical conjectures that I believe, and assuming quantum mechanics continues to be valid.”

Because different STEM cultures assign very different technical meanings to the term “asymptotic”, this “central question” remains meaningless until further definitions are given.

A Test Question  Today’s VLSI processors are 10^8++ times faster than the mechanical processors of Babbage and Turing, and they access 10^12++ times more memory. Shall we say that their net computational capacity is asymptotically the same?

Conclusion  Definitions of asymptotic computational capacity that conflate 19th century mechanical computers with 20th century electronic computaters are well-posed in a strictly mathematical sense (of course), and yet these same definitions have only marginal practical utility in scientific and engineering discussions … and in public discussions, insistence upon strict mathematical definitions can be grossly misleading.

For similar reasons, mathematical definitions that serve to strictly conflate VLSI computational performance with D-Wave computational performance have only marginal practical utility and in public discussions can be grossly misleading.