What is the quantum spin of a single electron in an (atomic) orbital?

[Curious Cat]

TL;DR Summary

I wrote a little (computer) program to fill in the atomic electron
configuration one electron at a time, for children, and it got me
thinking, about this.

What is the quantum spin of the valence electron in the silver atom in
the furnace in the Stern-Gerlach experiment?
. Up, down, at random, alternating, in a (quantum) superposition (of
both), or none? Does it even have/get one until it's measured/observed
/needed?
. Does the second electron, in an orbital, have to have the opposite
spin to start, with, or is it made to conform?

 

[gentzen]

In the furnace, the quantum spin of the valence electron of the silver atom should be random, no? There is no magnetic field inside the furnace relative to which there would be any reason that it should be up, or down.
(This can be described by using a density matrix instead of a wave function. One reason for using that description is that it makes it clear that there is no difference whether you assume that one half is up and the other half down, or that one half is left and the other half right, or that a quarter is respectively up, down, left, and right, or that the state is basically just totally random. ... But that goes into territory where only very patient children would be able to follow.)

The second electron in an orbital has the opposite spin of the first electron in the same orbital, otherwise it would be wrong to claim that both electrons are in the same orbital. Whether the orbital picture itself is fully appropriate is a separate question.
(Advanced level explanation: the orbitals which arise from the Hartree-Fock approximation cannot be fully appropriate. However, orbitals also arise from density functional theory computations, and there it is less clear whether they are appropriate or not, even so their exact interpretation remains unclear.)