Can anything spin in two directions at once, separately? If so, how? For instance, could an object spin sideways and up and down at the same time, separately?
The notion (and word) spin for subatomic particles is NOT the same as spin for objects such as a football. It's a purely mathematical idea, that unfortunately uses the same word as classical/normal objects. Elementary objects are, as far as is known, pointlike, so convential spinning does not apply to them.
A classical object, such as a football, can have it's spin resolved into perpendicular components, but I assume you mean two simultanously opposite directions, and no that's not possible.