Do I need hosting to register a domain?

No. Registering a domain and hosting a website are two completely separate services.

You can register a domain without having any hosting plan at all — for example if you just want to reserve a name for later, or if you only need it for email or to redirect to another site.

Without a hosting plan connected to it, the domain stays registered and valid, but visiting it will show a default page or an error, not a working site — there simply isn't any content associated with it yet.

If you later want to publish a site, you can always order a hosting plan (from EUReg or another provider) and connect it to the domain you already registered, without needing to re-register the domain.