This is all very confused. You have a 60 day tourist visa which you can extend one time for 30 days then you have to leave. This is not a multiple entry visa. You might be able to get a medical extension via the hospital but this is normally due to you being unable to travel. You should have applied for the Multiple Entry Tourist Visa (METV) if you want to be able to come and go, with this visa every time you enter during the 6 months it's valid for you get stamped in for 60 days (although you don't have to stay 60 days). There is also a specific Medical Treatment Non-O visa that allows you to enter for 90 days and which can be extended inside Thailand, that you can also get re-entry permits on.
she can get the Non-O inside Thailand but she requires a bank account and to transfer 800k baht into it from overseas before she can apply. It's currently pretty difficult to open a bank account on a tourist entry
I can assure you it is absolutely true. The airline staff announced on the intercom that tourists had to complete a TM6 form. I double checked with them as I found it odd too and was told I didn't need to complete one. It was post covid but before the TM6 was suspended. I don't travel out of Thailand that frequently so I'm very confident on when it was.