I know that graduate research students should not publish works without their professor's permission.
It's a little more complicated than that. Nobody needs permission to publish anything that they did entirely on their own. If you work collaboratively or as part of a team, then it's not 100% your work and you need your collaborators' permission. If you do not collaborate but work under someone else's supervision, then it's still polite to ask for advice/permission. (They might see it as more of a collaboration than you do, and in any case your submission may reflect on their supervision so it is reasonable to get their feedback first.)
Is this rule relevant to students in taught programs?
If you have no collaborators or supervisors in the research, then you can publish it however you'd like.
Can I submit my paper for credit in a course, without giving up ownership of the paper?
I don't know of any case where you give up rights to a paper by submitting it in a course, so ownership is probably not the relevant issue here. In the universities I'm familiar with, you can certainly submit for publication an essay that was written for a class assignment (most such papers would not be accepted for publication, but a few could be). On the other hand, you cannot get class credit for a paper that you previously wrote for another purpose. The precise rules in your case may differ, but if you plan to use this paper for a class you should investigate your university's rules.