Objects That Don’t Need Weights

The objects that are not going to be deformed don’t need to be weighted. You can just parent them to a bone. For example, the hair, cap, teeth, tongue (the tongue could deform, but in this case, to keep it simple, it will not), and the eyes (which should be parented to the H_eye bones, not the D_eye bones) would only need to be parented, as they won’t suffer from deformations by the effects of bones.

To parent them to a bone, just make sure you’re in Pose Mode in the armature. Select the object and, with Shift pressed, select the bone to which you want to parent them. Now press Ctrl + P to do the parenting, but instead of selecting the Automatic Weights option (the one you selected for the deformable options when adding an Armature modifier), pick the Bone option from the list.