1. What exactly goes inside of a coop? Is it just a wide open space and then some nesting boxes? I see other people's designs and am not sure why they designed certain things.
There are no hard and fast rules as to what goes in a coop. A lot of different things go into that decision. Climate, space, management techniques, number of chickens, flock make-up, many others. To thrive, a chicken needs food, water, protection from the elements, and protection from predators. You can provide those many different ways. As a minimum for the type of situation you have, I think your coop needs to provide a roost for the chickens so they can be safely locked up at night, safe from the elements and safe from predators. You probably want the nest box in the coop with that set-up. Food and water can go in or out of the coop. People do it both ways.
1a. What is a "roost"?
Chickens like to sleep as high as they can. It is instinctive as a way to be safer from predators, coming from originally sleeping in trees outside. If you put the roosts higher than anything else you do not want them sleeping in or on, they will sleep up there instead of sleeping in their poop. They do poop a lot while sleeping. Mainly what you are doing is trying to keep them from sleeping in the nest box. You don't want poopy eggs.
2. Why are nesting boxes in other coops always located halfway up the coop? Is it a problem to put them on the bottom like I have planned for mine?
Chickens don't care that much how high a nest box is. Some people put them on the floor. Some people put them at varying heights. Some considerations: A nest on the floor will get a lot of trash scratched into it. It will stay cleaner if you raise it a bit. A nest box that is high enough so the chickens can easily get under it will not count against the area of your coop available for the chickens. Chickens are basically ground dwelling birds. As far as space requirements, ground level is all that matters. If they are raised a bit but not high enough, the chickens will scratch the bedding under there but not go under there and scratch it back out, so you lose the space anyway. Some people, especially if they have a bad back, put the nest boxes up high so they don't have to bend over to get the eggs.
3. The coop and run are located on soil that was going to be used for a garden. It's not dirt. Is this an issue?
As long as it will drain and not stay wet, that should not be an issue. They will eat anything that dares sprout and will scratch dust bathing holes in it, so it will not look pretty. You might want to consider putting coarse sand in there if it stays wet to help it drain, but I really would not expect that to be an issue with it already raised like that.
4. Exactly how noisy are the chickens going to be? I have neighbors on the other side of the wall maybe 30 ft away.
Don't get a rooster. They will crow. The hens will cackle when they lay an egg and at other times, so your neighbors will hear them. I'd suggest talking to your neighbors ahead of time. Maybe offer them a few eggs?
5. I was planning on covering both the coop and the entire run with a roof. Is this necessary? I live in the Northeast and it snows. It will also block a lot of light. Is that an issue? There are trees around and it is shady.
I think covering it to keep the rain and snow out is a great idea. Don't forget about the snow load and make your roof plenty strong. You want to keep the coop and run dry, otherwise it will stink. A roof that diverts the water away from you run is a tremendous benefit. Most chickens don't like snow or wind. If you cover it and block it so the snow and wind does not blow in, it makes your run available year round. Chickens usually do well in cold weather. They are wearing a perfectly-fitting custom-made down coat after all. If you can keep the snow and wind out, you will probably find they are outside enjoying the run most of your winter days.
6. I read about people putting in removable floors. Is that sort of like a false floor? I can pull out the entire bottom of the coop to clean it and there would just be a gigantic hole in the bottom?
You have to plan your poop management. There are a lot of different strategies to deal with it. In that type of coop, I think it is almost a requirement that you have real good access to be able to open a door or a wall and be able to reach inside to scoop out the poop and bedding. Many people have a droppings board. That is a board under the roost to catch the droppings. They do poop a lot when on the roosts. If you remove the droppings, say by scraping them off the board into a pan, your bedding lasts a lot longer. Or some people use some sort of droppings pan that can slide out and be emptied. Lots of different strategies. Those droppings are great for a compost pile, by the way.
7. Should I make this coop multi-level? I'm not sure why.
I'm not sure why you would. You are really not going to increase the living area that much since they are ground dwelling birds, and you might make some areas that poop can build up and be hard to get to for cleaning. If you can overcome the cleaning access problems, having the roost up high with a droppings board to protect the feeder/waterer underneath may be a fairly efficient use of space.
8. How many windows should I put in this thing? Are the hens going to have an issue with too much light/too little light?
You might get a lot of different opinions on this one. I like to have at least one window so they can see to get to bed and get around in the coop. The nest box itself needs to be fairly dark as they are more comfortable laying in a secluded place, but they cannot see in the dark. They need enough light to be able to see. They don't need a whole lot of light but I think they need some. Be careful of the greenhouse effect. Even in New England you can cook your chickens in the summer if you are not careful, but the extra solar heat in winter is nice. Always tradeoffs.
9. I saw someone had hooked up some automatic door opener/closer. Do the birds need to be in the coop at night with a locked door or can they manage by themselves in the run? (Provided it is secure from animals)
It is harder to make a run predator-proof than a coop. I always recommend locking them up at night for safety. Even if you live in an urban area, you have a lot of predators such as dogs, raccoons, and possums. Probably several others. They have all night to figure out how to get into a run.
Sounds like you are in an urban/suburban area. Check your local laws about keeping chickens. Before you invest time, energy and money in chickens you need to know if you are legal.
I'd suggest you read these two articles. Pat is very good about these topics.
Pats Ventilation
https://www.backyardchickens.com/web/viewblog.php?id=1642-VENTILATION
Pats Winter Coop Temperatures
https://www.backyardchickens.com/web/viewblog.php?id=1642-winter-coop-temperatures
Why are many coops raised?
It makes the area under the coop available to the chickens. The more space chickens have the less likely you are to have problems with them.
It gives them shelter from rain and snow, plus a shady spot.
It gives you a protected place to feed and water them if you elect to feed and water outside the coop. The feed needs to be kept dry. This is often a great place for that. The water does not necessarily need to stay dry, but some people hang their waterers. This gives you a convenient place for the hook.
It raises it to a height that it is easier to clean without bending over or getting on your knees.
Remember, if you raise it, you will probably have to get under there sometime to retrieve eggs, collect a sick or injured chicken, get a feeder that they have scratched under there in the most inaccessible place. It is always something. I suggest you consider access to all points under there without having to crawl in chicken poop. Yours may be small enough it is not a problem if you can get to all sides, maybe with a rake. Just a suggestion.
Good luck!