After following the instructions on this very helpful post I was able to make eyelid animations work for my starter puppet. (How do you get Eyelids to animate for squinting and blinking? ). I've noticed that my puppets pupils warp as the eyes are closing but I would rather they stay the same size under the mask.
Bonus question: Is it possible to have eye lashes on a blinking puppet? It seems that without a blink layer, my fully closed eyes just disappear completely.
Eyes are one of the most complex parts of puppets - there are quite a few options. I wrote up a blog post that may be worth reading through: https://extra-ordinary.tv/2018/04/21/debugging-character-animator-eyess/
Eyes warping is usually because something has not been marked “independent”. There is also a “blink” layer you can use instead of eye warping, which can be combined with a cycle layers behavior for nice animated effects. You can certainly draw on eye lashes.
Character Animator does not understand what the pictures are - it just moves things relative to each other as you instruct it. Because most puppets are humans/animals, they name things with standard human body part names - but the software itself does not care what they are really.
If you control the pupils with the mouse, you can at least squint inwards. Just place the mouse between the eyes and the pupils will follow the mouse pointer.
For closed eyes with eyelashes I've always used an extra graphic.