Problem 1: Head-Turner-driven animation animates _into_ the head movement, but not back to Frontal in reverse.
What's happening is that the Head Turner triggers, say, Right Quarter, which starts the cycle in that group, holding on the last layer. All good so far. But when you go back to Frontal, from Head Turner's point of view, it just triggers the Frontal group. Cycle Layers says "hey, I'm not finished with my cycle (since On Trigger End is set to Let Cycle Finish), so it continues to trigger the Right Quarter group". So now _both_ Frontal and Right Quarter are being triggered at the same time. The way these conflicts are resolved is by triggering the frontmost layer/group. In your artwork, Frontal is in front of Right Quarter. So it wins, you never see the returning animation.
Solution: in Illustrator (or Ch, but better to make structural changes in the original artwork) move the Frontal group in front the profile groups. That way the conflict is resolved in favor of your animated groups, and you don't miss the animation.
Problem 2: Eyelids & pupils not working with combination of Cycle Layers and Head Turner.
If you select the top-level puppet and examine the Face > Views in the properties panel you'll see that it has found the 3 views (Frontal, Left Quarter, Right Quarter), and if you look at Face > Handles > Left Eye you'll see that it found the left eye in Frontal, Left Quarter/1, and Right Quarter/1 -- in other words, only the first frame of your turn animations. So the last frame isn't found, and isn't being controlled. The matching of handles for Face (and Eye Gaze) starts at the views, and only looks for one set of features per view. You might think you could just also tag the numbered groups each with Right Quarter, but layer tags don't match children within matched groups (would get confusing if it did). And if you _only_ tagged the numbered groups, it would work for Face, but they'd all get triggered by Head Turner at the same time, and you'd only see the frontmost group.
Solution: to force Face to find the inner numbered groups, select them all in the puppet panel and apply the Face behavior. This means you'll have a lot more Face behaviors applied, which is a downside for clutter, but an upside if you want individual control over the face in different views (e.g. in a profile view you might want to reduce eyebrow sensitivity because it's less accurate as you turn your head away). Note that when you apply a behavior to a group within a puppet, the same behavior applied at the top level will automatically not "find" handles and layers inside that group, so they won't get double-controlled.
The reason the mouths & auto-blinks were already working is that they don't care about views. Lip Sync matches on mouths within a group tagged Mouth Group. And Auto Blink just looks for layers tagged Left/Right Blink. So one way we could make this simpler is to have a new tag called Face Group, and have the Face behavior look for that group instead of the views (leaving those to Head Turner).