I am applying for an assisstant professor position, and asked to provide a CV, as well as a separate list of publications. Should I still keep a "publications" section in the CV, or refer to the separate list by writing something like "see enclosed list of publications"?
Yes, I would say so. A CV always should list publications.
It may be that for some reason to do with their search process, they find it convenient to have a separate document that only lists publications, rather than digging through the CV to find it. But that doesn't mean you should remove this list from the CV.
I think this depends on whether they put page limits on the documents. If they do, I would not waste pages on duplicating the publications but only list them in the separate publication document. A reasonable compromise could be to list a short selection of your most recent or most important publications in the CV and then provide the full publication list in the separate document.