There's no such thing as "AGI for the sake of AGI itself". AGI is synthetic and its goals are synthetic, it doesn't want anything that you didn't tell/construct it to want.
It's why there's a qualification to the term, because the old term "AI" was hijacked to mean the statistical mimicry we have today.
AGI by example: R. Daneel Olivaw, or the Minds in the Culture novels.
This is not correct. You can, in principle, construct a system with a set of random values, as but one proof by counterexample.
Another is an AI that is created as a blank slate learning agent with various feedback systems, and through interactions, it develops its own worldview. You didn't specifically construct it to want anything specific but it's "experiences" will shape it's future wants and thus behaviour.