Other way around you will get a 60 day visa exempt stamp, and you will be able to get another 30 days extension at your local immigration office for 1,900 baht, giving you a total of 90 days inside of Thailand.
Health insurance is not mandatory for your visit, but I would strongly recommend purchasing travel insurance just to be on the safe side, you never know.
You should be getting travel insurance before you leave home, as you don’t say what country you live in, but if you live in Australia for example you can purchase a 12 month travel insurance policy.