When you rigged this, did you manually create vertex groups or did you have blender do it’s auto-skin thing?
In either case, most of your problems now are related to vertex groups rather than parenting.
It looks like you did rig this manually judging from how clean your groups are
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If you go into the vertex group mini-window (in edit-mode), choose Head from the drop down menu, and click select, you will notice that all the vertices of your head are selected. Now deselect all the vertices of your mesh and do the same with the Neck6 group. The Head vertices get selected again. When you have a vertex that is in more than one vertex group, when you move the corresponding bones, that vertex ends up moving the average of where you moved the bones. For example, because your tongue vertices are in the Tongue* group and the Head group and the Neck6 group, whenever you move one of your tongue bones, the vertices will only move 1/3 the distance because it is partially held in place by the Head and Neck6 bones. To fix this, select the vertices of say Tongue3 ( for example). Then choose Head from the dropdown menu and click the Remove button. Do the same with Neck6. Now, the Tongue3 bone should move the Tongue3 vertices the right amount. Now just do the same with any other bone with the same problem (some being: Tongue2, Tongue1, Head, maybe more).
The Head vertices are also in multiple groups (like Neck6). This is why you are getting the deforming problem. Just remove the Head vertices from the Neck6 group and the Head won’t deform anymore.
Now, sometimes, having vertices in multiple vertex groups can be useful. For example, on the head, I would put some of the vertices that are between Neck6 and Head in both groups to get a smooth deform. If you just remove the Head vertices from the Neck6 group and rotate the Head bone, you will get a sharp angle/deform on the back of the head.
For the eyes, if you want them to follow the Head bone, parent the eye to the skeleton, only when the menu pops up, select bone. Then select Head out of the new menu that pops up, and the eye should be parented to that bone. You could also add dedicated bones for your eyes if you need to be able to turn them.
Rigging looks good so far, keep it up.
BTW: sorry for the really long answers, I’m not the best at explaining things in a concise manner
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