I believe I am right but my friend says that Clone comes back as 0/0 as therefore dies. Who is right?
You are correct.
Clone's ability reads
You may have Clone enter the battlefield as a copy of any creature on the battlefield.
This ability applies any time Clone enters the battlefield. The person who controls Clone (in this case you) gets to choose a creature to copy as it enters the battlefield. It will not die.
Can someone explain why Necroskitter allows the creatures to come back onto the field under their control? I assume that when someone plays a board wipe all the creatures does into the graveyard at the same time. Why is that not the case.
It is correct that all the creatures killed by a single effect go to the graveyard at the same time. But zone-change triggers such as Necroskitter's can "see" events that would trigger them even if the same effect is causing Necroskitter to leave the battlefield
603.10a Some zone-change triggers look back in time. These are leaves-the-battlefield abilities, abilities that trigger when a card leaves a graveyard, and abilities that trigger when an object that all players can see is put into a hand or library.
Example: Two creatures are on the battlefield along with an artifact that has the ability “Whenever a creature dies, you gain 1 life.” Someone plays a spell that destroys all artifacts, creatures, and enchantments. The artifact’s ability triggers twice, even though the artifact goes to its owner’s graveyard at the same time as the creatures.
Once the triggered ability has triggered and been placed on the stack it is entirely divorced from Necroskitter, and the fact that Necroskitter is in the graveyard is irrelevant to the resolution of the ability.