If you can get to a mall, then yes. But I have yet to see a sidewalk in Thailand that was pedestrian friendly, let alone wheelchair friendly unfortunately.
There is nothing stopping you from coming in on a tourist visa, or exempt and just having the surgery. You can set all that up before you get to Thailand.
I THINK it is the form you fill out for the hospital that puts you in their system. You will need your passport. I had a routine eye checkup at Bangkok Hospital, and although it was all booked online, I had to fill out the form once I physically arrived. I was told to come early because I had to register before any treatment starts. So, I would guess "the certificate" just says you are in the system as a patient??