I got braces when I was 21 or 22. Can't remember. I put $1500 down and then paid the ortho $200 a month after that. It wasn't a credit card or anything. I'm sure it depends on the ortho if they'd pull your other teeth or make you go else where. The total cost for mine was $5500 and I had them just under a year.
Mine was about $3000 which included everything (but I got a discount because I had braces previously as a teenager through the same orthodontist). There was no credit check, but everything had to be paid off before they would take the braces off. Some orthodontists have dentists in the same office, so you could try finding one like that.
Usually an orthodontist won't pull your teeth, you'd probably have to go to a regular dentist for that. My little sister has braces now and hers were going to cost around $4,000 total without insurance.