Thanks for the feedback! Roughly speaking, that all has to do with making e vanish later, so perhaps I need to revisit that section.
Quickly (cause I probably won't for a few days), (q//2)m can be seen as a form of error correction. You can check (either pen+paper or programmatically) that, provided |e| < q/4, if noisy_m = (q//2) m + e, then round(noisy_m / (q/4)) = m. So e vanishes because it is bounded (not uniform), + we encode m as (q//2)*m (i.e. in the "most significant bits" of the number).
Yes, many. I believe he's on the SPHINCS+ team (was standardized), Classic McCliece (round 3, not standardized), and NTRU_PRIME (round 3, passed over for Kyber). Perhaps more, but he has significant skin in the game.
LWE's hardness is based on SVP (ignoring issues of tightness, which isn't unique to FrodoKEM). The difference between FrodoKEM + Kyber/Saber isn't relying on SVP/not (they all essentially do), but is on relying on LWE over structured lattices or not.
At a very high level, all of the three rely on an n x n matrix at a certain point. The "structured lattice" schemes (Kyber/Saber) make structural assumptions about this matrix, say that each row is a cyclic shift of the previous row. This turns an O(n^2) object into an O(n) object, giving many performance improvements. The downside is that the additional structure can plausibly be used for attacks (but the best attacks ignore the structure, so this is a "potential issue", not a current issue).
Cyclotomic polynomials are incredibly standard in the field. The only researcher I know of who has issues with them is DJB, and there has not been significant advances in cryptanalysis due to usage of cyclotomics (with the exception of problems not used by NIST candidates, meaning the whole SOLIQUAY thing)
Quickly (cause I probably won't for a few days), (q//2)m can be seen as a form of error correction. You can check (either pen+paper or programmatically) that, provided |e| < q/4, if noisy_m = (q//2) m + e, then round(noisy_m / (q/4)) = m. So e vanishes because it is bounded (not uniform), + we encode m as (q//2)*m (i.e. in the "most significant bits" of the number).