That all sounds about right to me. I would like to point out though, the JET Programme isn't your only option. There are other English teaching organizations, as well as individual schools, that you can seek employment with. The JET Programme is just one of the better-known options out there, and most people have a decent experience with them. Also, you can apply with JET before you actually have the degree. You just have to have it by the time you are actually sent to Japan to work. A lot of people (including me) apply for JET while they're still in their senior year of college.
As for the visa, you can only renew it if you get another job. So on your last year of JET (or whatever teaching occupation you are in), you would want to be seeking new employment as soon as possible, that way you don't have to leave Japan. That, however, is a long ways off, both for you AND me
.