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Not stated, but basic common sense would dictate that Hope was always there to balance out the madness of the Great Evils, which we do have blatant evidence for existing alongside the Primoridials and Chaos, and that it affected them too.
Not how it works. Hope latched onto Kratos because he still held out that he would be freed from his nightmares, which allowed him to grow as powerful and as big as Ares.
Kratos didn't know, but his emotional state was such that it triggered the power to do its thing. The Great Evils and Hope all target the emotional aspect that the people/Gods undergo the most, in Zeus's case, Fear.
It depends if his emotional state is ready for unleashing it. But regardless, he still retains his godly powers. And we're never told if the Sliver of Hope that Kratos retained at the end of 3 would ever be usable for combat again.