I want to make a 'blink' shape key for this character. The eyebrows and eyelashes are on different UV maps and are different objects than the head. Is there a way to make the eyelashes and eyebrows follow the face's deformation? I.e. if I pushed the eyelids or lashes in in edit mode and made it a shape key for 'blink', can both lashes and eyelids move at the same time?
You should be able to do this by hooking the vertices of the eyelid to the eyelashes, so that the eyelash pulls them down, (which is what I think you are trying to do).
1) Select the portion of the eyelid you want to pull down, and create a vertex group for it in the vertex panel.
2) Select the eyelashes that will affect the eyelid, then select the main mesh (in this case, the body, if that is the mesh that the eyelid is part of).
3) With those two objects selected, press Ctrl+H, making sure you only have the part of the eyelid you want affected selected, and no other vertices.
4) Now you will notice that the eyelashes actually pull the vertices of the eyelid down with it when it moves, but it is probably going to look bad at first.
5) Select the main mesh, and go to the modifiers tab. Here you will see a modifier automatically added for the hooking object. You can adjust both the falloff type, radius, and the strength of the hook. This is for automatically adjusting the hook's influence proportionally. You also should select the vertex group you created earlier. This is for easy adjustment of the hook's influence. Using the settings in the modifier tab for the hook, adjust the strength and falloff until you get a value that nicely moves the eyelid.
Note: This method works best with an object that has not had any subdivision modifiers applied yet, because the adjustments will be much smoother.
Hope this helps!
You can definitely parent them to whatever head bone you may have, and they'll move with the head, or the eyelid bone, etc. though it may be a bit finicky once blinking, brow movements, and such come into play. It's generally considered better just to include and join them as part of the main rigged object for simple uses. There are probably a few methods I'm unaware of, similar to pinning cloth physics objects or something. People usually just put a big part of the UV tile aside for eyes and similar, buuuuut now-!
As of the most recent blender versions, use UDIM textures, which allow you to use many separate high poly textures (or low, you can mix and match) for single objects, as is standard in most high end 3D software now. https://www.youtube.com/watch?v=LR3Y9R2wmdI is a pretty good rundown!
Don't be afraid to put non-manifold things like eyelashes or eyeballs in the same object to be controlled by the same rig or skeleton, especially if it's a simpler one. Choosing whether or not a mesh should be joined to its supposed parent object borders on the philosophies of zen and whether you're comfortable doing it in this use case. you'll find a lot of models doing it one way or the other in various cases, so get used to philosophizing and agonizing a little over it.
a lot of people also use hair particles for those, because you can use procedural materials, and as a 3d model, it will then technically have no functional resolution limits.