Nicola,
You do not have to wash chickens. Some people do, especially if they are going to show them, but it does not effect a chicken's health if you do not wash them. And if you do wash them, you need to keep them out of a draft until they dry or they could chill. You will need to treat them for mites and lice. The dust baths the chickens take is their way to combat mites and lice.
I don't know how cold it gets where you live. Chickens can handle (-) 7 degrees C without a problem. If it gets colder than that, they need to have a well-sheltered place to go. By well-sheltered, I mean a place out of any wind or drafts. Heat is not necessary unless it gets colder than maybe (-) 20 C. As you said your back yard is mostly weeds and dusty dirt, you may be where it is hot and you need a lot of ventilation and don't have to worry about the cold.
For what you plan, the 1.2 meter x 2.4 meter coop is about the right size for the 3 chickens you want. That gives about 10 square feet per bird. Your back yard is plenty big enough for the chickens. Most people would love to have over 400 square feet for 3 chickens to run in. As to the height of the coop, the chickens would be happy with 1 meter clear over them. You probably want the height to be comfortable for you as you will have to go in to clean it. Shoveling out chicken poop while you are on your knees would not be much fun.
I'm a little concerned when you say your mother has to have the birds free-range. Is this because she thinks they can feed themselves? Chickens need to be able to eat all day, not just after school. You are going to have to feed them when they are in the coop. They will get a lot of food from the greens and bugs in your back yard and they can eat a lot of table scraps, but they have to have a balanced diet. If I am wrong about this assumption, please do not take offense.
As far as keeping your coop costs down, you can check what people are throwing out on trash day before the dustmen pick it up. Talk to friends, ministers, school teachers and other trusted adults and tell them what materials you need. They may be able to help. Look for buildings that are falling apart. You might get materials for just tearing it down. And be flexible in your plans and try to use what you find. The chickens are not worried about the coop being pretty. It does need to be strong to keep predators out.
As you go along, you will have questions. Just post them on this site and you will get a lot of help. Not all of us here agree on everything, but there are a lot of great people at this forum and they will try to help you.
You seem to have a good spirit and a great attitude. Taking on the responsibility for the lives of three animals is big and I admire you for even considering it.