You'd probably be better off sticking to weight painting for this model. Bone influences allocate vertexes to bones using a geometric limit system - ie. if a vertex falls within the inner sphere it will move with the bone, if it is beyond the outer sphere it is unaffected, and if it falls between the inner and outer spheres it is partially affected depending on distance. This can have advantages over weight painting in that you can't miss vertexes, which is fairly easy to do with weight painting, but it can be difficult to prevent unwanted overlapping of areas. For example, if using influences on a skinned hand it would be easy to have the outer influence of one finger influencing vertices from a neighbouring finger, producing unwanted results. With weight painting, you define how much vertices are influenced irrespective of their distance from the bone, providing far more control, but perhaps less consistency.
The eyeball problem is common when animating models based on toys, which generally over-exaggerate eye size. One possibility is to use a section of a sphere only, with the bone pivot point set to where the centre of the eyeball would be, even if it is outside the head. This still requires room around the eye socket for the eye ball section to move into, but possibly the eyeball section could be morphed to distort as required, so as not to project outside the head when moved to an extremity. The main advantage bones have is they can animate rotation, which generally looks better for eye movement than the linear motion morphs are limited to.
By the way, lovely model, and good organic modelling is something that takes time and patience to master, which is why I have barely attempted it! Some call it an art... I call it sheer torture!