But are you still considered to be a student when you are a post doc? Do post docs get "salary" during the summer, or do you have to find a summer job?
No, I don't think you're considered a "student" as a post-doc, exactly, but it's still a "training" position so it's not quite faculty. There are also a lot of differences between fields, types of post-docs, etc. so it's hard to say anything "in general."
Someone can correct me if I'm wrong, but I believe most post-docs receive 12-month salaries.