Hi Evan, the "eye" that you are referring to is Adobe's way of telling you if that particular layer is visible or hidden. In AI, PS, AE is matters if it is selected or not because it obviously displays artwork on that layer or hides it. In CA the artwork can be shown even when it is marked as hidden (or hidden even if marked as visible) because of CA's ability to use triggers and swap sets.
So, in the example in your screenshot of Alamasol above, the "Ah", "Uh" and "W-Oo" mouth shapes are made of several layers that are rigged with a behavior (that's that little Lego looking icon with the gear in it) - probably to cycle layer animate. The "Ah" only has layer 2 visible, but the behavior will cycle through all the layers in the "Ah" group when it is triggered. Similarly, whenever the "W-Oo" viseme is required, CA will display the sequence of three layers (which are all marked as visible.)
Now the only time this gets a bit tricky in CA is when you use PS or AI to edit a puppet that is already open already working with in CA. When you change a puppet in PS or AI, CA automatically updates your puppet file. So sometimes if you have a puppet that is already rigged, and you change the visibility of layers in PS or AI, it turns them off completely in CA, when it updates. Just go into CA's rigging mode and turn them back on.
I hope this makes sense. Good luck and happy animating!
This is clarifying. Thank you for your helpful response.