Can I operate a paid language teaching website while on a retirement visa in Thailand?

To do this requires a big commitment, opening a company and starting a business is a lot of work here as it is anywhere in the world. You don't need a lawyer or law firm to open it, it's pretty simple, really, but it's a commitment. There are a million websites on how to set up a Thai company, here's one of the better ones.

Yes... But with a catch 😉 you can have a website that teaches Thai for a fee and have a retirement visa.... But you can't be in Thailand because you don't have work permit 😏 see how that works? 😂 So in your situation living in Thailand the answer is no.

The problem isn't changing visas. The problem is getting the required work permit. You'll either have to be hired by someone else, or start your own business, and that gets expensive just to meet the work permit requirements of having at least bt2million in registered capital, and hiring 4 Thai employees. For someone else to hire you to teach you must have a four year university degree. It's not easy to do.

You may be able to obtain a work permit married to Thai.

You are on a Non OA Extension of stay based on retirement that doesn't permit work of any kind whether paid or otherwise. The clue is in "retirement".

It would be unwise to jeopardise that and in particular post your intentions on a public forum that even the IO can follow.

No. You are not allowed to do any type of work including volunteer or free on that visa.

If someone reports you and they will. You will be deported. Welcome to Thailand like many other countries they protect their work force.

You cannot do anything with a retirement visa even if its free