Can a student write a paper and send it to a professor review?
Sure, why not?
But let me suggest a better plan. What you need is a faculty mentor. If you have a favored topic that you are working on, then first simply go and speak about it with a knowledgable professor about it in person. It will be extremely helpful to you to have a faculty mentor in your mathematical work. You can explain your idea and result and discuss it with him or her. Please try to listen to what the professor says about your idea. In the best situation, he may be able to guide you by providing some ideas that will help you to extend or improve your results. (In other circumstances, you may learn that the result is already known, or trivial or wrong in some way, which would be extremely important for you to know if it is true.)
In the best circumstance, the professor may encourage you to continue on the project and write a paper. In this case, most professors would also be willing to review the final draft, or even to supervise an undergraduate research experience based on the topic. Usually writing a mathematics paper involves many more drafts and revisions than novices expect, particularly when the author does not have much experience with mathematical writing. The help provided by a faculty mentor will enable you to bring your work to a higher level.
Yes, this has been done and in fact there is an award for undergraduate mathematics research the Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student
Permit me to draw attention to a relatively new journal, Involve—A Journal of Mathematics. Snippets from About the Journal:
"Involve is dedicated to showcasing and encouraging high quality mathematical research involving students (at all levels). ... Submissions in all mathematical areas are encouraged. However, each manuscript should include a minimum of 1/3 student authorship. ... Involve is a publication in between the extremes of purely undergraduate research journals, which in general are aimed at undergraduate audiences, and mainstream research journals."
I would guess that nearly all of the papers that have appeared in its first nine issues reflect work supervised by a faculty mentor, and a good percentage are coauthored with that faculty member (and not infrequently with more than one student).