Therefore, it is conceptually wrong to think of [Lorentz transformations] as acting on the points of the spacetime manifold M
It's definitely wrong to apply the Lorentz transformation to coordinates on some arbitrary pseudo-Riemannian manifold, as the output will be meaningless. If the manifold is flat in some region, and your coordinates on that region are Minkowskian, then it's not wrong and it's sometimes useful.
If "conceptually wrong" means "pedagogically ill-advised" then I think it's conceptually wrong to base your understanding of special or general relativity on Lorentz transformations at all. We don't understand Euclidean geometry in terms of Cartesian coordinate transformations because we have an evolved intuitive sense of how it works that doesn't require coordinates. It's better to try to adapt that intuition to spacetime. As a consequence of its intrinsic structure, certain mappings of points to points in the Euclidean plane (resp. spacetime) take valid compass-and-straightedge constructions (resp. systems evolving in a way that's permitted by the laws of physics) to other valid constructions (resp. other valid histories). As a special case of that, if you define a certain type of coordinate system, and write one of your mappings in terms of those coordinates, it may have the form of a Cartesian/Lorentz transformation. But the universe doesn't care about coordinates or Lorentz transformations as such, just about the intrinsic structure of the thing that you're trying to mathematically describe.
Notably, the laws of physics appear to be local, and don't treat even a proton, much less a spaceship, as a single conceptual unit, so any Lorentz transformation that acts outside of a differential neighborhood is in some sense going beyond the scope of the laws of physics. Nonlocal Lorentz transformations work (when they do) only "by accident." But they're still useful, and you should still use them; you just shouldn't assume that the universe uses them.
People usually use $x^\mu \to x'^\mu$ as the formula for the Lorentz transformations without mentioning that this is in any way improper.
They're either talking about Minkowski coordinates on a flat region of spacetime or about a vector in a tangent space. They could plausibly use $x$ for either one.
How to think about the Lorentz invariance of laws, e.g. of the Maxwell equations?
It's really just rotational invariance. There are probably a lot of different ways that you could understand or formalize the symmetries of an arbitrary Riemannian manifold with an ordinary positive-definite metric, and all of those carry over to pseudo-Riemannian manifolds. The latter only seem more complicated because we don't have an evolved intuition for mixed signatures like we do for the +++ signature.
(Actually mixed signatures are theoretically more complicated in some ways – e.g. the group of rotations isn't compact – but I think that for the purposes of your question it doesn't matter.)
The transformation $x^\mu \to x'^\mu$ also seems to be used in the derivation of Noether's theorems.
I don't know anything about most of Noether's theorems, but the famous one called "Noether's theorem" doesn't depend on Lorentz symmetry; it works also in Newtonian mechanics for example.
If Lorentz transformations take place in the tangent space and translations take place in the spacetime, then how does it make sense to talk about the Poincaré group that encompasses them all?
In general it doesn't. The Poincaré group is the isometry group of Minkowski spacetime. A de Sitter or AdS or FLRW spacetime has a different isometry group. A realistic spacetime like large-scale FLRW with a bunch of randomly placed stars has no nontrivial isometries at all. Since the laws of physics are local, there is no physically meaningful difference between the highly symmetric spacetimes and the nonsymmetric ones.