As long a you are white you should have no problem getting a job in Chengdu.
Yup, pretty much. New regulations require that only expats from native-English speaking countries are eligible to work in English teaching. If you are white and from America I do not forsee any obstacles in your path.
As far as getting a school to pay for your flight to China: you can do that, but they will want to sign a contract. And from what I’ve observed over the years (I’m not a teacher), the jobs you find here in Chengdu tend to be a lot better than those you’ll find online. So although you would have to pay for your flight up front, it might be better in the long run (if you’re planning on staying for a year or more) to get here and then start looking for jobs.