By law and by SUNY Board of Trustees regulations, students own the copyright of any work they create as part of their studies. They give tacit permission for their work to be copied, distributed, and remixed only as much as necessary to complete class interactions and grading. Faculty must ask and receive permission from a student to use any piece of their work for any purpose outside the learning and grading activities of the course for which it was made.
Students in group projects may be separate copyright owners of individual pieces of work, or joint authors of a collective work.
Students have extremely broad ability under Fair Use to incorporate other copyrighted works into their classwork - much broader than would be permissible in any other environment. For example, if a student copies a diagram into a class discussion, they must cite it, but they do not need to get copyright permission.
If the work will ever be accessible outside the course (used in the community, put in a repository, presented at a conference, etc.) the student must either remove the external copyrighted content, or get permission for it.
Students who research and write with faculty must be credited as contributors or authors according to the same rules as apply to faculty colleagues. Academic status is irrelevant to copyright law.