How come John never tried to sway Spades Slick over to their side like Karkat did? Or how did Karkat even figure out to try to? It just seems awfully fortuitous for the trolls that things panned out that way.
It wasn't fortuitous. It was Aradia -- the dead Aries girl. She fused her Sprite with a Frog statue, which in turn caused the Black Queen to take on frog-like qualities. The Black Queen, who despises all things amphibian-like and found the form insulting, removed the ring that gave her all of her powers (and her chimera-esque form), and locked it up in a vault.
Jack Noir/Spades Slick, who wished to overthrow the royalty and take the ring of power for himself, contacted Karkat and together formulated a plan to break into the vault and steal the ring. At the end of the Troll's game, he apparently ended up as one of the Exiles.
Meanwhile, by the time John had even heard of Jack Noir, Noir had long-since exploded the Black Queen, taken her power ring, and was on his way to wrinklefuck all of Skaia.