Can I get a Non-O visa after arriving in Thailand on a tourist visa or visa exemption, and extend it for retirement purposes?

Understand that you do not need any special ED visa in order to go to school. You can do it on your retirement visa once you get it. You can even go to school on a tourist visa, as quite a few do with Thai cooking schools. Concentrate on your retirement visa first. You can do pretty much anything you want on it, except work.

in order to apply for the "change of visa type" to a 90-days Non-Imm-ED visa, you will need all the relevant paperwork from the school ready to show Immigration. There is no guarantee that Immigration will accept it . . Check the requirements here, Your situation is in number 3, just click on it

The O in non-O is "other"

The non-O is for both retirement, marriage, and a few other things. You just apply for it based on what you're eligible for.

To apply for the non-O based on being 50 or older in Thailand you'll need the 800,000 baht in a Thai bank account along with the other requirements. You should go to the immigration office you'll be using ahead of time and ask them for their handout as requirements vary by office and sometimes they change from year to year or with changes in staff.