Ive only been checked on Sukhumvit in BKK , never anywhere else and I have traveled all over Thailand but after that I never go out without a copy of my passport since a foreigner really has no rights in Thailand.
I always carry a photocopy of my passport and the page with my immigration stamp and have never had a problem when they checked me but I once saw the police yelling at a foreigner they had checked and they took him away in handcuffs.
Flora, its actually pretty simple, you get 30 days when you arrive and if you want to stay longer just go to the nearest immigration office before the 30 days are up and request and extension and you will have no problems. Do not try to overstay without a proper visa or you could find yourself in very serious trouble if they catch you and the police are always stopping foreigners walking on the streets and checking their visa so your chances of getting caught are high.
You can get the 30 day arrival visa and then go to the immigration office before it expires and pay for an extension which if I remember correctly will cost you 1900 baht.
No you must buy your own travel insurance and although thai hospitals are much cheaper than in the USA travel insurance is still a much better deal. You can buy it online or at any AAA office and will cost about $140