Assuming that one has begun a PhD program at a university which allows a student to choose an advisor after his/her first year in the program and the student has narrowed down on a couple of potential advisors and has taken AT LEAST one course with each of his potential advisors:
What should the student take to the first meet? A rough research idea, summary of the professor's important papers or otherwise?
Note: I have made 5 separate questions as suggested (As opposed to the single one which I started off with)
For an initial meeting, I would take an general idea of the kinds of things you'd like to do in a particular advisor's group, a sense of what the advisor works on—you don't necessarily have to have read papers in advance, and an open mind. That said, you'll make a better impression if you have read a few recent papers, and have a few intelligent questions to ask. It will pique the advisor's interest, and show your sincerity.
As for coming in with ideas for a research project, this depends a lot on how the project will be funded. If you have your own fellowship and can design your own project, then you should definitely have an idea or two ready. On the other hand, if you will be funded by an existing research grant within the group, your choice of projects will be more constrained.
One final note—I would say that it's definitely not required that you take a course with an advisor before considering them as a research advisor. If that were the case, I would have been on a completely different career path than the one I'm on now, as I'd be working for my fourth-choice advisor (who was the only one I took a class with before choosing a research group!).