Kishi has already answered the first two questions:
1.What happened to the ball of rock Nagato created after he died?
As you can see, the rock started collapsing once Naruto went back to normal:
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2. Wouldn't mankind need food or water under the Moons Eye plan?
The Moon Eye plan requires you to use Infinite Tsuykomi. To use that you need to be the Jinchuriki of the Juubi:
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Only Madara needs to be Jinchuriki:
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Now Madara posesses Hashirama's DNA which was created from the Gedo Mazo, and we know that you can survive without food and water with it:
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Since Madara is the only person that needs to be alive for the Moon Eye Plan to work, and he can survive without food and water because of Zetsu, mankind does not need food and water for the Moon Eye Plan.
Now I don't have an answer for the last two questions, I'm afraid, but I'll speculate.
3. If the Genjutsu of the Moons Eye Plan is cast on the moon what happens when the sun comes up.
If Madara becomes the Jinchuriki of the Juubi, he will probably be as powerful as the Sage of Six Paths. Now we know the Sage of Six Paths made the Moon:
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Maybe Madara might make another Moon with Chibaku Tensei which will stay close enough to the Earth that we can see it at all times of the day.
4. Will Kabuto die from hunger in Izanami?
I don't know.