The cosmos may, and probably does, extend more than 13 billion light years from us. Some parts are receding from us at what seems to be faster than light speed - possible because they aren’t moving, space is expanding. We can never know anything about this region of the universe. It isn’t quite an event horizon in the black hole sense, but it is similar.
Sure. But natural explanations are deterministic at the macro level and probabilistic at the quantum level. We can derive laws and equations to describe natural things. But consider the plight of Flatland scientist trying to describe the motion of this sphere within their universe. (or the point that is the intersection of the sphere and their plane.) The sphere appears, then disappears, then shows up someplace else. In 3d we can use the laws of motion to describe its movement, but in 2d it is totally random. If the plane is curved, the sphere can even move faster than light as it travels along the plane. So from the Flatlanders view it is violating all sorts of physical laws.
Okay, the extra dimensions of string theory are not supernatural. Neither is the third dimension in my analogy. I used it to examine a similar case, where the third dimension stands in for the supernatural, but is not the supernatural. If I spoke of supernatural events for the flatlanders, we’d be no further.
It boils down to what you mean by natural. If anything that projects onto our natural world is natural by definition, you’ve defined away the supernatural, since whenever the supernatural impacts our world it becomes natural. That is like the Flatlanders calling the sphere two dimensional because it has two dimensional effects. Or perhaps you define natural as being subject to natural law. I contend that the sphere, though subject to 3d (supernatural) law is not subject to 2d natural law in any meaningful sense - unless you consider popping up at random intervals and moving faster than light a law.
I’m not ruling out that we could start hypothesizing supernatural law if we ever found evidence of the supernatural, just as a genius flatlander might be able to posit a third dimension in which the movements of the sphere made sense. All those theists who say that god has to send people to hell are doing just that. We won’t know until we get evidence.
So, if supernatural events ever happen, with an effect on the natural world, I agree that science can and should study them. They should of course first try to explain them naturally, but it might turn out that a separate, supernatural explanation is better than an absurd natural one. And all this would be tentative, as usual. It all boils down to that the best reason to reject the supernatural is lack of evidence, not that it cannot exist in principle. Skeptical scientists who investigate (and reject) supernatural evidence do a great service, since otherwise the field would be left to the fruitcakes. And I include magicians among scientists here, since they are the best observers.