Actually, you are wrong: While you think this is "clearly impossible" there is nothing clear here and it is completely unknown whether this superposition of states is possible or not. On quantum-level it is, but quantum-effects apparently do not scale up.
So, have you ever experienced a cat being alive and dead at once? Can you ever, really, imagine this happening with anything you experience directly? Just because you can say the words doesn't mean you understand something. This is one of the most fundamental tenets of formal logic - you cannot occupy two mutually exclusive states of existence at the same time. It cannot be raining and not raining. A cat cannot be alive and dead.
Nevertheless, QM makes it abundantly clear that this happens all the time in the realm of very small things, and it is easy to describe thought experiments (such as the feline predicament) that inherently tie the state of large, classical things to microscopic, quantum things. So we can make the impossible happen (or so Schrodinger suggests). The whole thing was conceived as an objection to quantum mechanics.
The one thing that is clear is that for sentient beings above a certain level, the superposition is impossible, because if it were possible the person in it would immediately collapse it. For inanimate objects it is not a problem at all having a superposition like this, you just move the time of the random event and adjust its nature. For the cat, it depends, see my (mostly satirical) statements above.
That's not clear in the slightest. Researchers have been devising lots of experiments to demonstrate quantum behavior in progressively larger objects. First was photons, then electrons, eventually atoms and molecules, with the current target being viruses and then potentially larger single-celled organisms. There's nothing in a fundamental physics sense that makes a living thing more special than ordinary matter - we're all just collections of atoms, after all. What's more, we don't know that sentience causes collapse, or anything on that level.
We DO know that quantum objects behave as particles when the paths they've traveled are knowable (at least in principle), but as waves if the information about the path is unknowable. We've got variants of this for different kinds of particle behavior, with the fundamental idea being that particles exist as probability waves until they are measured and observed in a certain location. What part sentience plays in this is anybody's guess - if it truly does play a part, that means something about our minds changes the state of the universe itself, and if you don't find that preposterous and world-changing, you haven't begun to understand quantum mechanics.
I do not know what you find "unsettling" here though. As long as superposition does not apply to beings on the sentience level of a human, everything is fine for you. Really, get over not everything being as it looks.
I'll let some of the people that developed QM speak for me here:
"Those who are not shocked when they first come across quantum theory cannot possibly have understood it" - NIELS BOHR
"Quantum mechanics makes absolutely no sense" - ROGER PENROSE
"If you are not completely confused by quantum mechanics, you do not understand it" - JOHN WHEELER
"I think I can safely say that nobody understands quantum mechanics" - RICHARD FEYNMAN
And finally, from the man himself: "I do not like it, and I am sorry I ever had anything to do with it." - ERWIN SCHRÖDINGER