Okay, I'll take an initial crack at this-
A dowel rod, if made of wood, would cause a LOT of damage to the eye as it wedged into said area. At the least there would be tearing of the sclera. At worst the eye would go pop (as in, pop open). That's nay good. I don't know if you could get it to pop out in this scenario. Perhaps if the rod was then jammed over...
Read up on the guy who had a metal rod go through his eye(?) and up into his forebrain, WALKED to the hospital, and lived. But, the rod went through his emotional (?) center and he was never the same (Someone will chime in with his name).
The forebrain is not critical for life. I'm not sure a dowel rod would penetrate bone (depends on the material) and into the brain, especially if it only went 3 inches. My guess is it would jam the optic nerve of the eye in question and possibly pull on the other optic nerve- thus causing issues in the other eye.
Also in that area are sinuses, but I haven't studied the human skull. Go have a look at the skull, in detail, and decide what your rod is doing (where it's going)
Does that help?