You are missing the point. The EU makes laws that govern - and at least try to - protect its citizens. If a document on the archive is created by a European citizen, then it is under EU law. That's why every company in the world right now that deals with European citizens is working on supporting GDPR. That also applies here.
Not quite. The EU might want that but it gets into jurisdiction.
The EU cannot enforce its law on entities that are entirely US based. It can only enforce it on non-EU sites if that site has some sort of business that’s within the EU (like offices or employees).
The EU can say that some businesses are so uncompliant with GDPR that they're not able to be used by EU companies. It seems weird to chose to limit your market just because you don't want to protect user data.
Not GDPR-specific, but France.com had its web domain seized by France recently[1]. It was a private US-based business (not a squatter) that the government of France had actually cooperated with for years, until they suddenly decided it violated French trademark law and seized the domain. The domain itself had been in one person's possession since 1994.
Enforcement of national laws is very much a thing across borders, so private businesses outside the EU are right to be apprehensive about what is going to happen as GDPR enforcement ramps up.
