The Schrödinger's cat with a non-quantum source of random choice As far as I understand the "classical" thought experiment with Schrödinger's cat, it uses a Geiger counter with a very small grain of radioactive material. As I understand it, Schrödinger's idea was that such a small grain of radioactive material is a quantum object, and thanks to the whole device, its quantum properties are transferred to the macro level. More precisely, the superposition (|Undecayed particle>+|Decayed particle>) turns into a superposition (|Cat is alive>+|Cat is dead>). What if in this experiment, instead of a radioactive substance, we use a “classical” source of a random event, for example, a mechanism which tosses a coin? From the formal point of view, the superposition of a living and a dead cat will disappear?
 A: Yes.
In classical statistics, you can have various probabilities for certain events that take place, but they can affect each other only sequentially e.g. if event x takes place with a certain probability, then a second event y might be affected by the result of x. In essence, probabilities are the fundamental quantities in and of themselves, living on an abstract space that pictorially would look like a straight line between 0 and 1, with each point being any real number between them. There you can only add or multiply probabilities between themselves.
In quantum mechanics, because probabilities are derived from complex probability amplitudes that exist on a Hilbert vector space, they can mix and co-exist in superposition, which gives rise to this (intentionally, by design) absurd assertion that a cat can be both alive and dead before an observation/measurement takes place.
An intuitive way to understand this is the usual paradigm that particles behave like waves, or perhaps more aptly that their probability amplitudes behave like waves, which then allows for wave-like qualities such as superposition. Since a coin (or its probability) are not wave-like, the concept of superposition isn't possible.
A: The lesson I would take from the Cat Paradox has nothing to do with superposed cats: it is that the meaning of a theory is what we can say about the future or past state of the universe given certain knowledge about the current state of the universe: a theory means what its model predicts, and a model is a statement of pure logic whose meaning is uniquely contained in its formal mathematical expression.
According to Schrodinger, the Cat Paradox points out that the intuitive/descriptive mode we use for thinking about quantum phenomena in terms of "blurred" values is often useful but not likely true. It is the psi-function itself that faithfully represents reality, not the physicist's attempt to visualize it.
If you want to treat the coin flip as an amalgamation of innumerable indeterminate quantum events which is practically impossible to model but in principle could be described by a psi-function, and you assume that one characteristic of the psi-function is that it gives rise to a 50/50 split of heads and tails states, then you can in-principle describe the cat's superposed states as being groupable into one collection of superposed alive states and one collection of superposed dead states, each collection having equal probability. But what does this mean? It means do the math and make a prediction... which is obviously identical to the classical prediction in this case. When we open the box, we'll either find a live cat with a heads-up coin, or a dead cat with a tails-up coin, and if we do the experiment a million times, half of them will be alive cats with heads-up coins.

[An alpha decay from a nucleus] is described, if one wants to explain
intuitively, as a spherical wave that continuously emanates in all
directions and that impinges continuously on a surrounding luminescent
screen over its full expanse. The screen however does not show a more
or less constant uniform glow, but rather lights up at one instant at
one spot[...].
One can even set up quite ridiculous cases. [The cat experiment is described.] The psi-function of the entire system
would express this by having in it the living and dead cat (pardon the
expression) mixed or smeared out in equal parts.
It is typical of [macroscopic states contingent upon events with quantum indeterminacy] that an indeterminacy originally
restricted to the atomic domain becomes transformed into macroscopic
indeterminacy, which can then be resolved by direct observation. That
prevents us from so naively accepting as valid a "blurred model" for
representing reality. In itself it would not embody anything unclear
or contradictory. There is a difference between a shaky or
out-of-focus photograph and a snapshot of clouds and fog banks.
[...]
[T]he indeterminacy is not even an actual blurring, for there are
always cases where an easily executed observation provides the missing
knowledge. So what is left? From this very hard dilemma the reigning
doctrine rescues itself by having recourse to epistemology. We are
told that no distinction is to be made between the state of a natural
object and what I know about it, or perhaps better, what I can know
about it if I go to some trouble. Actually - so they say - there is
intrinsically only awareness, observation, measurement. If through
them I have procured at a given moment the best knowledge of the state
of the physical object that is possibly attainable in accord with
natural laws, then I can turn aside as meaningless any further
questioning about the "actual state", inasmuch as I am convinced that
no further observation can extend my knowledge of it - at least, not
without an equivalent diminution in some other respect (namely by
changing the state).

(Schrodinger, translated by Trimmer, American Philosophical
Society, 124, 323-38)
