Am I right in understanding that individual atoms have a magnetic field? If so, does it have anything to do with charge and spin?

"Spin" is by definition the magnetic moment of the individual particle. So it has everything to do with spin.

A material which is magnetized has the spins of the electrons lined up so they add up to a larger total magnetic field. In a non-magnetized material, the spins are randomly oriented so they cancel each other out.

In the simplest terms imagine a hydrogen atom with an electron going around the nucleus.

Imagine that instead of orbiting in all planes it was like the earth where the electron only orbited in a single plane.

You would have a current ( electron) moving in a circular path ( a coil) generating a magnetic field.

Now this is an oversimplification but it shows that if there is any assymetry in the orbits of electrons the magnetic fields will not cancel to zero.

Yes---speaking simply, the movement of the electron generates a magnetic moment and field.

https://phys.org/news/2014-05-exploring-magnetism-...