As awful as that is, the design of those dumpsters will always lead to this. To put the bag in, you must hold the lid open well above your head (and higher than many people can reach) while holding a heavy bag of trash, then lift it even higher to get it in. If you are smaller than average (e.g. a child), physically disabled, or just not an able-bodied adult, that becomes impossible
I boil with salt and a bit of vinegar in the pot. When I’m done, there is a noticeable white coating on the pot, presumably calcium from the egg shells.
So yes, it gets washed.