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.

Stopped Equius from possibly killing a bunch of people by being his Moirail, got Aradia, possibly the most helpful member in their session into the game, AND most importantly fucking allowed Fefeta to exist, poor, sweet, dear, precious Fefeta.

Well let’s see, there’s the

Vast Honk (Sound LE makes when he enters a universe)

Vast Glub (Sound Glubglub makes that kills all the trolls)

Vast Croak (Sound the Genesis Frog makes when he’s created)

And then there’s

Red Miles (Unavoidable attack causes by the Queens’ ring)

Yellow Yard (The Yard Stick Hussie places between the two 4th walls that John and Jade traveled between)

Black Inches (Snowman’s whip)

I think that’s all of them

She died, came back as a sprite, exploded, became a pirate, found some treasure, her crew left her, her pre-scratch ancestral counterpart turned out to be a dick, she realized that she was also a dick, then learned about the power of friendship and made out with a fish-lady.

It just wears off.  

Players have 24 hours to beat the Black King once he starts the Reckoning.

As far as I’m aware, the average amount of time before he starts the Reckoning is usually a few weeks like Karkat’s group had, although it might differ by session.

The Beta kids’ Reckoning started super early (Less than a day in) due to interference by Bec Noir.

Everyone in a successful session, yes