If soemoen sues you for infringement of a design copyright, it would be a valid defense that the plaintiff had in fact taken the design elements from your prior work, or indeed from the work of a third party, and so had no valid copyright. Proving this may be easy or hard, depending on the specifics of the situation.
Registration will prove that your work existed and you claimed it as original on a specific date. It would not be a perfect defense: if the other party claimed that you had copied prior to that date an infringement action could proceed.
The effect of registration is somewhat different in US law and the law of various EU countries. In the US, for one thing, one must register before bringing an infringement suit, and the date of registration may affect the availability of statutory damages, and of an award of the costs and fees for bringing suit.
Note that the protection offered by a design copyright is limited: copyright will never protect an ide3a or concept, and so only the specific expression of a design can be protected by copyright. What this means will vary in particular cases.
Specific artwork or logos will each carry its own copyright, and if they are original creations of yours, you can register them. (These would not be design copyrights, but ordinary copyrights.) You can also obtain a single registration for the website as a whole.
It is very unusual for one who copies a work, or elements of that work, to then sue the original creator for infringement.
Given the existence of archive services, it is often fairly easy to prove how a site looked on a particular date. Ensuring that your site has been archived as soon as possible would make it easier to show that it was not copied from a later work. Some services will archive a particular site on request.