If a work enters the public domain, can someone "privatize" it by trying to copyright/trademark/etc. the same work without changes?
Under US copyright law, there are two statutory provisions that correspond to "being in the public domain". The first is that US government works are not protected by copyright, therefore they are "in the public domain". The second is that copyright has a specific duration, so if a work is "too old" in that defined sense, it is also "in the public domain" and is not protected by copyright. In either event, copyright is a right held by the author, and "is a US government work" is "is too old" is an immutable fact that cannot be revoked.
The third possibility is via a "public domain license", whereby an author declares that anybody is free to use their work – like an ordinary license, without the "in consideration of X I permit you to Y" verbiage. A bare license is generally revocable, ergo you can enter my bar without my explicit permission, but I can revoke that license to enter. It has not been definitively settled by the courts whether a bare copyright license is revocable, instead licenses usually include some quid pro quo condition that makes the license be a contract. There are also statutory provisions allowing for the eventual termination of a license, so this third path is not a definitive entry into the public domain.
Regardless of the "public domain" status of a work, copyright only arises from creating a work, therefore if a person creates nothing, there is nothing protected by copyright. You cannot try to / fail to copyright a work, because the right known as "copyright" is automatic with the creation of the work. Registration of copyright is completely separate act, and does not create a copyright (though it is a necessary act in order to sue a person for copyright infringement).
Copyright is unrecoverable
You don't gain copyright for publishing a work. You only gain copyright when you make a work. Once copyright lapses because the author is dead long enough or the corporate work was published long enough ago, that's it. Nobody can have copyright in this one work.
Trademark does not privatize a work
Trademark is to identify the origin of a product that is used in commerce. For example, Coca Cola. Some trademarks share names with IP, but they only extend to the trademark, not the works: There's a Conan holding that owns multiple Conan trademarks, but the works by Robert E. Howard are still out of copyright. You can re-publish the Conan stories, but can't infringe on those trademarks.
Separate works have separate copyright.
I mentioned Robert E. Howard and Conan for a reason: while the stories by Howard are out of copyright and public domain as are the characters with it, the adaptions like the 1982 and 1984 films are still under copyright. Those are separate works, they have separate copyright.