You can buy a cheap departing airticket to ensure you are not turned away on arrival..which ya wont be but pays to be safe. I cant remember name of the business that sells them...onward ticket or something like that. Its like 20 bucks.
Yes you'll get 60 days mate plus the option of extending it by another 30 days for an extra 1900 baht at Thai Immigration when you get here. So 90 days total is possible.
Yeah its a piece of a4 paper with about 6 diseases written on it. Any Doctor will tick it and sign it. Pretty straightforward. You get it from the Thai embassy in your country along with the other visa paperwork to fill in.
Police background check yup. I think every health insurance company wants to know about ya health. I use AXA Thailand. They are very good and not too expensive.