Well, let's take your questions into account one by one.
1. Mao would likely be replaced by a high-up in the CCP.
2. Without the GLF, China would prosper. The GLF was a massive waste. The CR doesn't happen at all, since that was Mao's attempt at knocking down the government to gain power.
3. Mao would be beloved by everyone in China, and likely remembered well in the West. Keep in mind, Mao didn't become an idiotic buffoon in the minds of the Chinese government until the late 50s.
4. Will Mao's cult of personality be as developed? The PRC would make use of it, but it wouldn't approach the levels OTL. He'd still have a huge founding-father mythos about him, though.
5. Authoritarian? In such a world, he would be seen as downright libertarian. He was genuinely interested in establishing village-level democracy - it's just that he was Mao, and as such, a fucking asshole.
6. Who would take up the mantle? Either Zhou Enlai, Zhu De, Liu Shaoqi, or one of the other ones. I doubt Deng would get the job, sadly.
7. Does China fix itself faster? Probably. If only because Mao isn't competing with Stalin to see who can get a higher killcount, China would do much better. Furthermore, the leadership's more moderate/less radical, meaning they'd be more willing to embrace capitalism (although then again, without the trauma of the Cultural Revolution, Chinese leadership might not get pushed to capitalism).
8. Long-term fate? Better, along with the rest of China.