It should be all the kids. 

There’s no basis for this in canon but I like to believe that anyone that walks through the end game door has some ownership to the new universe.

To wake up his dreamself that was unconscious 

crumpledpaper-inyour-bag

replied to your

post

:

Do all Time player have the ability to reset a…

why do they need to posses a needlekind?

You need Needlekind in order to scratch your Scratch construct.

no

That can’t be true because Karkat and Kanaya were frog breeding for weeks.  If they had left the Incipisphere after a day they couldn’t have done that.

If they possess needlekind and their session hadn’t been reset prior then yes.

The humans are still playing Sburb and the trolls are helping them, yes.

No there are no other kind of death that leads God Tier’s to their eternal slumber.

To answer the other part of your question I quote Doc Scratch when Rose:

Broadly, mysteriously, and according to the case of the individual.
One may be killed by opposing a corrupt adversary and die for a just cause, as through martyrdom, for instance. This would be heroic.
Or one may be subject to corruption, and slain by a hero. This would be just.

No.

John’s was to free the fireflies, which he did by removing his planet of oil

Rose’s is to play the rain and bring life back to her planet’s ocean

Jade’s was to stroke the forge and breed the frogs

Dave’s hasn’t been stated directly in canon, it could relate to the Forge or the legendary blade Caledfwlch.

The Alpha kids’ has not been stated.

They learn their quests by learning their planet’s lore and by speaking with their denizen.