The chicken with its head cut off saying extends beyond the south. I grew up hearing it in the Midwest. Since chickens do run around when their heads are cut off, I think anywhere with chickens would end up with this phrase.

The Always Sunny thing is more about her looks. When they call her dumb, they tend to just call her dumb/stupid/etc.

I would imagine the Dodo is a big reason for the birdbrain insult meaning dumb. Since Dodo itself can be used to mean dumb, and since they weren't afraid of humans they were easy targets and hunted to extinction. That may say more about humans than the Dodo.