We have 3 goats, 1 turkey, and 9 chickens (3 of them are roosters). Everyone lives together, night and day, and we don't have any problems with that.
I did some research on-line before my daughter got her goats, and I came up with my own version of a goat-proof chicken feeder. It hurt my brain to design this, but I just knew I could do it! Most of it is made out of an old cupboard.
With the exception of a few minor adjustments, I've used this successfully for almost 2 years.
The top opens up to fill feeder. I made two access doors for the chickens. The one on the left/front had to be blocked off because one of the goats could get his face in just far enough to stick his long tongue up there to reach some feed.
The other door (hard to see) is on the right on the side. The hole in the front is where the turkey sticks her head in to eat. The feeder is up on cement blocks so she can reach it.
Since then, my daughter requested I make a chicken-proof goat feeder because they were laying eggs in the hay. I recently had one little bantam that decided to be defiant and didn't even care that the goats were eating while she layed her egg! The chickens cannot get into the fresh hay, but she layed her egg in the tray underneath.
With all the spots in that coop that are available to lay eggs, almost all choose to lay eggs underneath that tray. We even had a broody hen hatch 2 babies under there.
I realize someone could get stepped on someday, but the chickens are FAST, and it's very common to see them run, or just walk, under the goats. And we occasionally find a goat laying down with a chicken laying on its back.