Right way of requesting research project from a professor as a non-affiliate
It is acceptable, but not the best strategy. Getting the attention of a top professor is like fishing a very picky fish: you need a good bait. In my experience, it is easier to approach them by offering something rather than by asking for something. Also, the approach should be genuine. This is, you approach them because you want to work on something that might fit his/her research interests not just because you want to see his/her name tied to yours in a paper.
IDEAS: Professors salivate when they see an interesting idea.
Just follow all the etiquette rules. Be polite and respectful as you would be with any other person. Also, try to be precise, they are busy people.


More importantly: Publications are not strictly necessary to be admitted to a decent PhD.

If you have some project in mind, write it and send it to the Professor in question. Tell him/her that you are interested in working on such project and that given his/her research interests, you think that the project would be greatly benefited with his/her ideas and that you would appreciate collaborating with him/her on it.