I "lock down" the camera because I have a tendency to forget I'm not in perspective view and whiz off on a tangent, which really messes things up if I'm doing multiple shots in a render.

Select the camera

Go to the parameters tab.

Click the lock icon on the x, y, and z translate and rotate slides.

No need to spend any time point eyes at the camera or adjusting the camera for eyes, Just create a null at the same location as the camera using the steps you said , point the eyes at the null, and move the null or the character a bit as needed.

To move the null, you select it in the scene tab. You can use the arrows on screen or go to the parameters tab, general, and move it there. You can enter spcific xyz coords by clicking on the number and entering a different one.

Primitives are on the create tab.  You can enter dimensions and number of sides when you create the primitive.