How would one go about making an eye blinking animation for a character and have it play, but it doesn’t stop all other actions? Because when I do this using actuators, no matter the priority (other actions and the blink animation), it just shows up as suspended animation and doesn’t properly play.
How can I make a character blink like in most 3D games? Layers does not help. Also, its just the eyes itself, that are animated.
Make the blinking animation with Shape-Keys.
Prevent the animation from stopping when other animations are played by marking the animations with individual Prioritys and/or Layers.
I played around with it, and your solution worked! I had to go through a bit of trial and error to see what priorities and layers I had to do. Rather than blinking via shape keys, I kept it as armature, and it still worked!
Glad it helped.
Just throwing this out there, if you don’t know how to use shapekeys I would highly recommend learning how, because it was amazing when I learned how 
(It took me years to even try, and it was great)