We have wall feeders inside the coop, and really like them for the ease in filling and also the floor space they save.
They are a little on the small side, so we have two for 23 birds. We probably fill them a couple times a week, but our chickens also get our table scraps, and free range part time.
They don't have covers, but it doesn't seem to make a difference. They're too narrow and close to the wall for the chickens to roost on. Our coop is pretty critter-tight, but even when the doors are open all day we don't seem to have much of an issue with anything but chickens getting to the food.
If you wanted covers for them, it probably would be an easy matter to cut a piece of plywood and hinge it to the wall to come down over them, but we haven't found a need for it yet.