Where should I put their water in the coop then?I just read through the entire thread.
You do not have enough space in that little coop for 2 birds let alone one and trying to integrate your single bird is going to be (already is) a nightmare.
You need more space and a properly ventilated coop, with roosts. You only need a single nest box for three birds. And it should not be taking up valuable floor space.
You need (as has been mentioned more than once) a proper balanced feed for the flock. I highly recommend putting the entire flock on Flock Raiser, All Flock or a non-medicated chick starter. Any complete feed that offers 18-20% protein.
It is very difficult to maintain a flock in snow country without proper space, either in a very large coop or in a coop with an attached, predator proof, solid roofed run.
If you want to have healthy happy birds, you are going to need to provide them with better housing and diet. I think all your problems will be resolved once you've met their basic needs.
As to the integration, they need lots of SPACE, things to hide behind, things to perch on, things to dig around in and multiple food and water stations. Your single bird is squawking and complaining because she doesn't want to be alone. She should be inside a large wire dog crate in the coop/run area with the others so they can adjust slowly to each other but it doesn't look like you have enough room for that. You need to have 4 sq ft of space per bird in the coop and at least 12 sq ft (preferably much more) per bird in the run. The more space the better. Always.
It appears that their run is where a gutter downspout discharges. That isn't good. Their area needs to be kept dry. The cup with water in it nailed to the inside of the coop door should be removed. It's a spill risk and a humidifier that will encourage frostbite when it starts to get really cold.
Can you stand back further from your setup and take some more pictures and post them?
Where exactly are you located? You can update your profile with that general information so it is always displayed with your posts.