A contributed paper session is a session where anyone can submit a talk (rather than having to be specially invited). The criteria for approval are usually not selective and mostly just involve a check that your topic is appropriate for the session / conference.
If I register for a contributed paper session, can I register with the paper that has already been published before in a journal (I am the author of the paper)?
Yes, it is fine to give a talk on work that is already published. You should include a citation of your paper in your presentation.
Can I use my paper that has not been published yet?
Yes, this is also fine. If you have a preprint that is publicly available (e.g. on arXiv), it would be good to give a link in your presentation. You can even give a talk on work for which the paper is not yet finished, but this is a bit risky, as someone in the audience may decide to work on it, and get their paper submitted before yours.
Does the title/abstract of such a talk need to be the same as the title/abstract of the paper the talk is based on?
Not necessarily, but it's a good idea if the titles are similar. That way, someone who attends the talk or sees the abstract will have an easier time finding the corresponding paper. For the abstract, your abstract should describe what you will actually speak about, and so the abstract from your paper may not be directly suitable. For instance, your paper probably contains lots of details that you won't have time to address in your talk.
If I am not the only author of the paper, but I am the only speaker at the conference, should I list only myself as the author of the abstract?
For the Joint Meetings, all the authors of the paper should be listed as authors in the abstract. (For projects not yet published, list everyone who is expected to be an author of the paper when it is finished.) There is an option to indicate which author is actually giving the talk.
Other conferences might have different practices. If coauthors are not listed explicitly as authors, they should be acknowledged within the abstract, i.e. "This is joint work with Name and Name."