What is the quantum spin of a single electron in an (atomic) orbital? I wrote a little (computer) program to fill in the atomic electron
configuration, one electron at a time:
What (quantum) spin do I give a single electron in an orbital?
Up, down, random, alternating, (in) a superposition (of both), or none!?
Does it even have/get one before it's measured/observed,
in a particular axis/plane? I now suspect not.
 A: 
What (quantum) spin do I give a single electron in an orbital? Up, down, random, alternating, (in) a superposition (of both), or none!?

The spins of all electrons in an atom depend on the spin of the protons and neutrons in the atomic nucleus and among themselves. To make it more visual, electrons are also small (bar) magnets and the spin is aligned parallel to this. The orientation of the magnets (the spins) in the atom is subject to entanglements.
The best known entanglement is formulated by the Pauli exclusion principle. If two electrons occupy the same energetic states, then their spin (magnetic dipole) is oriented in opposite directions.
However, the formulation up and down is misleading (but perfectly adequate for most cases). In reality, a single atom can rotate freely in space and all its spins rotate with it. Not so in chemical compounds or even more extreme in Bose-Einstein condensates. In the first case, the bonds between the atoms of the chemical compound determine the position of the spins. In the case of the BEC, many atoms or their spins aka magnetic moments are entangled with each other.

Does it even have/get one before it's measured/observed, in a particular axis/plane? I now suspect not.

For free atoms it is random distributed. As soon as you start measuring, you will influence the atom. It is no longer free and depending on the measurement method, you will align the atom more or less with its summary spin aka magnetic moment.
A: It is useful to see the hydrogen atom available orbitals:

There are three quantum numbers of the wavefunctions and they define each orbital, including the spin (the m quantum number) the orbitals do not mix  spins, the spin quantum value defines an orbital.
After the hostile comment ( missed it before Triaticus comment), let me spell it out.
The electron orbitals have specific spin assigned to  them, there is an up spin orbital and the electron there will always have its spin up, and there is a down orbital, where the electron that occupies it will always have spin down. If there are two electrons (an ion) they will each stay in its orbital.

What (quantum) spin do I give a single electron in an orbital?

The spin assigned to the orbital by the solutions of the quantum mechanical equation. The example above is for hydrogen orbitals and wave functions, but in order to have an orbital there should be the analogous solution of the quantum mechanical equation.
