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Some of it depends on where you live and your climate. There are plenty of people that have wire walls for one or more sides of the coop. Chickens need protection from predators and the worst of the elements. How much protection from the elements depends on your specific weather. Consider the worst days, not the best.
Chickens can normally handle cold better than heat. They wear a down coat year round. A huge advantage of a wire walled coop is that you can get great ventilation and get rid of that dangerous hot air. Along the Gulf Coast, an open walled coop is tremendous. Many people make them work much further north. If you can position it so the worst of the wind does not blow directly on them when they roost, it can work. Maybe you need the roosts in that protected corner more than the nest boxes?
Nest boxes don’t need to be totally closed in. Sometimes they will hide a nest from you under a bush or even in a clump of high grass. What I’d suggest for you in an open coop is to put them so rain can’t hit them and get the bedding and eggs wet. You could even do that by building a nest box on the wire side of the coop.
If they are locked in there, the grass won’t last long. It will soon be just dirt. Some people do OK with a dirt floor in the coop but I think most of us use bedding of some type. With an open coop like that, you don’t want to use something that will stay wet. Their poop stinks when it gets wet and the bedding can get moldy. Mold can make them sick or kill them. I’d suggest you leave it dirt or maybe think about covering it with sand and see how it goes. Sand is not a bad thing at all.
How much you need to clean it depends on your specific situation. My coop is pretty large, enclosed so it stays dry, and they don’t spend a lot of time in it. I use wood shavings on the floor and put up a droppings board. They poop a lot when they roost and droppings can build up pretty thick under the roosts. I catch the worst of those on a board under the roost and scrape that off about once a week to put in my compost pile. I haven’t cleaned my coop out in over three years. I do have a feeder in there so they are always raking it for me, scratching around to try to find spilled feed. Some people scrape the dropping board daily and/or clean their coop weekly. Some use sand and rake it regularly to filter out the poop. We all have different situations. I can’t tell you what will or will not work for you.
Good luck and welcome to the adventure.