In the US, registration is not required for copyright, but it changes the amount that you can win in court over a copyright violation. (This may be a violation of the Berne Convention but the only way to find out would be someone not in the US, pushing for a copyright case in the US, not getting the results they wanted, and then taking it to an international venue.) Without registration, you can only sue for provable damages; for registered works, you can push for more than that.
For what kind of rights: Copyright in the US is a limited-term economic monopoly. It covers copying, public performance, derivatives (like translations; what else counts as "derivative" is a game for lawyers) and a few ancillary rights. In the US, copyright does not cover "moral rights" that exist in Europe and some other places. Music recordings have their own special (nightmare) section of the law.
Unpublished works are copyrighted - this is relevant for things like "an author's rough draft of a novel"; someone can't grab it and publish without permission. Copyright exists/is granted the moment a thing is put in "fixed form." An unpublished game in your cloud storage is copyrighted; if someone hacks your account and steals it, they can't (legally) publish it without your consent.
There are rules about updates and new editions that would apply to registration of video games. I don't know the details, just that books have been doing expanded/updated editions for ages.
(But also: yeah, this is not a good venue for discussing the philosophy of copyright.)