How does the IK know to stop turning the character's head when the target goes behind him? Is this a property of the rig, and/or a Unity setting?
The fingers aren't posed to exactly hold the sphere, and some fingers go through the sphere. It seems like a rather complex process to have to reposition all of the fingers. What would be the normal procedure to get this to be more accurate?
You can adjust the muscle range of motion within the rig settings. So it's built into Unity here: https://docs.unity3d.com/Manual/MuscleDefinitions.html. As far as repositioning the fingers, it's not an easy process. The simplest solution is to create animations specifically for the object they're holding and mask out everything else with an avatar mask. The more complex solution is using a custom IK setup to control the individual fingers or use something like Final IK.