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As far as I know, the probability of a quantum object being in a certain position depends on the wave function value for each position. That raises a question: Is this probability strictly greater than 0 for all points? If I place an electron in a box, it can be anywhere on the box, or anywhere on the universe?

For example, there is always a small possibility of finding a value however far from the mean, while that is not the case for a triangular distribution.

Also, slightly related: Is this property maintained when studying classical objects? Is there any possibility, even if unimaginably small, that all of the particles of a cat will simply move somewhere else at the same time, "teleporting" it?

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You answer your own question in your first paragraph:

the probability of a quantum object being in a certain position depends on the wave function value for each position

If the value of $|\psi|^2$ is greater than zero there is a probability of detecting the particle. Whether this is the case depends on the potential. For example if the particle is in an infinite potential well then $\psi$ is zero outside the box and there is no probability of detecting the particle outside the box. If the box is not an infinitely deep potential well then there is a probability of detecting the particle outside the box, though this probability rapidly falls to such small values that it is indistinguishable from zero.

To take a real example, for an electron in the ground state of a hydrogen atom the wavefunction is non-zero everywhere so there is in principle a probability of detecting the electron anywhere in the universe.

Re your follow-up question, the issue of to what extent QM applies to macroscopic objects has already been extensively addressed. A quick and far from exhaustive search finds:

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