If I wanted to copyright the name of a PRODUCT, CAN i DO THAT THROUGH THE COPYRIGHT.GOV WEBSITE?

Product and business names, and indeed any short phrases, are not copyrightable. Copyright is limited to works of authorship, which generally means books, plays, photographs, software and similar items. You may protect the name of a product under the law of trademark. You should engage the services of a trademark attorney to determine whether the name you have in mind is available and non-infringing, and regarding the steps to be taken to protect your mark.