It is unfortunately the case that many sources refer to antiunitary operators as just "operators" (e.g., the "CPT operator") even though in almost all other contexts "operator" implies linearity rather than antilinearity.

This paper claims to describe gauge theories with a mass gap but without confinement. I don't know how to square that with the above discussion. arxiv.org/abs/1702.05988

After seeing answers to this question that attribute the stability of L4 and L5 to the Coriolis force despite the instability of the potential at those points, readers may wonder whether L1, L2, and L3 are stable despite the saddle shape of the potential at those points. An answer addressing these issues is here: 363801/7911.

Worth emphasizing: if something depends on the representation, physically this means it depends on the total angular momentum, e.g., the anticommutators will be different for spin-1/2 vs. spin-1 particles.

Yes, they are distinct but strongly related. In general, the intuitively preferred quasiclassical (sub)systems, such as the center-of-mass (CoM) of macroscopic objects, cannot be permanently preferred. The bound atoms that today compose a (say) chair were widely separated in the distant past and will be again in the distant future. However, insofar as the chair is bound together robustly during some time interval, we expect wavefunction branches to be approximate eigenstates of the position and momentum of the chair's CoM.

FYI, Hartle's paper, and the frequentist approach to deriving the Born rule from within Many Worlds in general, has been criticized for relying on the Hilbert space inner-product norm to define the $N\to\infty$ limit, which implicitly bakes in the answer. See for instance Kent [Int. J. Mod. Phys. A 5, 1745 (1990), arXiv:gr-qc/9703089] and Squires [Phys. Lett. A 145, 67 (1990)].

Thanks, you nailed the key idea. I have handled the representation multiplicity issue in my answer here.

@Quantum Mechanic : I am asking for which pairs $s_1$ and $s_2$ there can exist states $|\psi_1\rangle$ and $|\psi_1\rangle$ such that $\langle\psi_1 | \sigma_n |\psi_2\rangle$ can be non-zero; I've edited the answer to make that clear. Not sure what the correct terminology is, but I was searching for ladder operators that changed the eigenvalue of $S^2$, whereas I would describe what you're talking about as "ladder operators for $S^z$" (which, as you say, keep you in a fixed eigenspace of $S^2$, such as the symmetric subspace $s=N/2$).