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If the coop has not been used for a year, you are not worried about mites and lice. They cannot live that long without chickens. It is unfortunate that the creepy crawlies will be killed by disinfecting. Chickens see them as extra protein and really enjoy finding them and playing keepaway with them. But some diseases can last a year or more without chickens being present, so a good disinfecting is called for. I'd hose it down good, spray it with the 10% bleach solution, then let it dry in as much sunshine as I could. We've all got our different standards but this would be enough for me as long as I soaked it fairly well with the bleach solution.
In the normal terminology on this forum, the box is the coop and the fenced in area is the run. That looks plenty big enough for 4 hens. Since you mention mongoose, I assume you are in Hawaii. The normal recommendation is for 4 square feet of space in the coop along with 10 square feet in the run for each full sized chicken. Chickens are basically ground-dwelling birds so height really doesn't matter. There are a lot of assumptions that go into this and it is intended to cover everyone from Anchorage Alaska to Miami Florida, from Perth Australia to Aberdeen Scotland. One of the big assumptions is that they will be locked in the coop for extended times when they are awake. If you are in Hawaii, the weather should not force you to lock them in there too often so you could get by with a smaller coop if necessary. I think you will be OK with 4 hens in that.
Whether you let them out is a personal decision. There is always a predator risk but we all have different risk tolerance and different predator situations. With only 4 hens, the loss of one would be pretty big, but many people do it.
Planting something for shade is a great idea, but they will eat practically anything green they can get to. If you do plant something, I suggest planting it back far enough that they cannot reach the roots or main trunk, maybe a bush instead of a vine.
I'm not sure what gauge wire you have on that coop. Light gauge chicken wire will not stop a dog. I'm not sure about mongoose. I understand the variety of mongoose in Hawaii is a daytime predator and that they can get through some pretty small holes. I think mongoose make your situation a bit different than a lot of us and certainly different than mine. Again, it is a personal decision based on risk tolerance, but a strategy many of us use is to let the chickens in the run (or free range) during the day and lock them in a very predator proof coop at night when most predators are active. To keep things from digging under the fence, I prefer the apron someone else mentioned. You take about 18" to 24" of wire, lay it horizontal on the ground, attach it to the bottom of your run, then either leave it laying on top of the ground or maybe just put it under the sod. The idea is that the predator goes up next to the fence, starts to dig and cannot get through, and does not realize it can back up. I think this is more effective than trying to bury it straight down and a whole lot easier to install.
As far as the area inside the run, leave the grass. They will destroy that pretty quickly but they will get grit form the ground and will love dust bathing in it. They like to scratch in the dirt. Another big thing about keeping healthy chickens is to keep the run and coop dry. Disease can really cause a problem in wet manure or dirt. Any organic ground covering will hold moisture. Sand or sand over pea gravel works really well. As long as it has a place to drain to, it drains really well and is pretty easy to rake if you need to.
Inside the coop, you have some choices. You can use many things inside the nest. Straw and wood shavings are probably the most popular, but some people use shredded newspaper, Spanish moss, pieces of carpet, or who knows what else. All you need is something they will not scratch out as they are making the nest and that will cushion the egg so it will not break when they lay it. It helps if it is easy to remove and clean.
The floor of the coop is a bit different. They poop a lot while on the roosts at night. You need to remove the poop before it builds up enough to be damp. The moisture can cause disease problems and the ammonia given off by damp manure can cause them respiratory problems. Again, there are a lot of different strategies to deal with that. Most involve having something on the floor like wood shavings or sand to absorb the moisture, then clean it out before it starts to stink. Your nose will pretty quickly tell you when you need to clean it. Others scrape the poop out on a regular basis. Either way, it is a great addition to your compost heap. You will need a way to dispose of the poop and whatever floor bedding you use.
You'll get a lot of conflicting advice on this forum. That is because we all have different set-ups and different experiences. Chickens are pretty adaptable and tough. As long as you provide food, water, and adequate shelter from the elements and predators, they will do well. I suggest you real all the responces and pick out what you think best suits your situation. And, again,
