Hi,
So I'm also autistic and have sorted the uni support out
So with the proof of disability you should have a copy of the ASD report (your parents might have this if you haven't seen this)
If you do have the report this is enough to send to the university and DSA and don't need a letter from the doctor if you have the original diagnosis paperwork. If you don't have the paper work you should be able to either ask for a copy of it from the doctors as it will be useful to have your own copy to bring to uni incase you want to apply for things you need it for or you can ask the doctor to fill in the form confirming the diagnosis.
They normally say the diagnosis report is better as it lists your struggles but if you were diagnoised at 3 I'm not sure how accurate it would be
For the exam adjustments I had a meeting with the university advisor and we spoke about different support etc and included in that was about exam support and what I had in the past, I told her and she didn't ask for any proof but got all the adjustments again so they should be able to give you them just from asking for them
For the ensuite you would need to apply for DSA but I'm not sure if they are able to get an ensuite for autism (I wasn't), it is normally for a physcial disability but it might be possible.I'm not sure what proof you would be able to get for it though. I also had a problem using the bathrooms at school but for me it was the hand dryers and other sensory aspects but at uni there is 3 toilets for 6 people and they are in a room by themsleves so like one at home, but I don't know if you would have a problem with this as well