What is written on the visa doesn't matter. The visa was used as soon as you entered Thailand. Now what matters is the stamp they put into your passport when you crossed into Thailand. That is what tells you how long you are allowed to stay. That stamp would only be for 60 days. And you can get a single 30-day extension giving 90 days total. But if you're already almost at 90 days and haven't gotten an extension, you would be on overstay.
What visa did you buy from the embassy before you traveled to Thailand? If you indeed had a 90 day visa then you most likely had a non-O visa and there is no 30 day extension available for that. The 30 day extension is only available if you're on a tourist entry which means tourist visa or visa exempt entry.
A non-O visa can only get a 1 year extension after meeting all the requirements.
There's no such thing as an approved clinic. There is no master list of approved and not approved clinic. Each embassy sets their own requirements and decides what does and doesn't qualify for DTV. It's more about what the treatment is going to be than what clinic you go to. But each embassy makes the actual decision.
But as far as what type or duration of class will qualify, that is going to be up to each embassy to decide. Some report wanting a few weeks of classes, some want a month of classes, or some want 3-6 months of classes in order to qualify.
Seems like the medical option is actually an easier way to qualify for DTV, as some have reported getting the visa with only a single dentist appointment.
1) Easiest and guaranteed way, find an agent and pay about 5000 baht to help you open the account.
2) More difficult and not guaranteed way, get a residence certificate from your immigration office, and then start going into bank branches and asking. Maybe after 50 or so you might get lucky and find one that will let you open an account, or they might let you open an account if you buy their insurance for about 5000 baht.