I don’t think you should submit paperwork for each, no, and you shouldn’t do deposits either. You and your son can visit the school and talk with them and then decide on which one. You might investigate accommodations but honestly, they are not optional: they are mandated by law.
Once the acceptance has gone through, you can meet with disabilities formally (they will want to meet with him actually), submit paperwork, and talk about accommodations. A single room is a common accommodation. The bathroom will be a harder issue. Think about social pros and cons of a single too. A small suite might be ideal.
Be aware that disabilities offices aren’t always geared to helping students so much as filtering requests to keep the integrity of the curriculum and avoid " undue administrative and financial burdens" for the school. They can arrange things through housing (or direct you), and will provide letters for your son to take to professors. Sometimes accommodations have to be negotiated and that is when other personnel are needed.
We found deans and medical personnel to be key in applying pressure to professors, to be honest, and it took a year or so to figure out the system and how to make it work. The main message is, don’t go by the disabiltiies office only. Talk to a dean or other administrator and find out who else can help. Sometimes there are extra advisors. If there is a health center, they should be aware of your son’s medical issues and a PCP there will be important.
For documentation, I suggest you write the letter for providers. They don’t resent this! The docs or therapists or whoever can doctor it up but you need to research and network and figure out the list of accommodations needed. You don’t need documentation for each accommodation if they are listed under a diagnosis that covers them all.
Feel free to PM me. I assure you that this can be worked out and your son can thrive but it is not easy.
Pick a school before doing anything at all complicated. Once he accepts the offer of admission people will be cooperative and you will have the right to have all the meetings he needs.
One other thing: I am a firm believer that kids should not choose schools based on anything to do with disabilities. That really goes against the civil rights the ADA law intended to insure. We had to trailblaze a bit but every school should be able to understand reasonabe, well-documented needs for accommodations and should provide them.
You will find you cannot just drop him off and may occasionally have to nudge things a bit, but only in the beginning!