Can I directly obtain the 60-day COVID extension for visa-exempt entries in Thailand, or must I apply for a 30-day extension first?

depends on where you go to do the extensions. When I did my first one (30 days) I asked if I could get the 60 days covid extension and they said no I had to get the the entitled 30 days first. At the time I read on some post that some people were able to get the 60 days covid extension right away. I went to Samui Immigration. FYI the last day to be able to get a 60 days covid extension is May 30th. Unless things change and they extend this date as they did many times before. But if I were you, I would go before the 30th just to be safe. Good luck. As for TM30. I've never been asked for it.