Regurgitating the teacher's password is a matter of mental process, and you have nowhere near the required level of evidence to make that judgement here.

As for radioactive decay, I'm not clear what you require of MWI here. The un-decayed state has amplitude which gradually diminishes, leaking into other states. When you look in a cat box, you become entangled with it.

If the states resulting from death at different times are distinguishable, then you can go ahead and distinguish them, and there's your answer (or, if it could be done in principle but we're not clever enough, then the answer is 'I don't know', but for reasons that don't really have bearing on the question).

Where it really gets interesting is if the states resulting from cat-death are quantum-identical. Then it's exactly like asking, in a diffraction-grating experiment, 'Which slit did the photon go through?'. The answer is either 'mu', or 'all of them', depending on your taste in rejecting questions. The final result is the weighted sum of all of the possible times of death, and no one of them is correct.

Note that for this identical case to apply, nothing inside the box gets to be able to tell the time (see note), which pretty much rules out its being an actual cat.

So... If you find Schrödinger's cat dead, then it will have had a (reasonably) definite time of death, which you can determine only limited by your forensic skills.

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Note: The issue is that of cramming time-differentiating states into one final state. The only way you can remove information like that is to put it somewhere else. If you have a common state that the cat falls into from a variety of others, then the radiation from the cat's decays into this common state encodes this information. It will be lost to entropy, but that just falls under the aegis of 'we're not clever enough to get it back out' again, and isn't philosophically interesting.