Here an overview of all Korean sets/decks, focused on Pikachu releases and my personal collection. 
In terms of set releases, Korean went from the Base Set, to the ADV Expansion Pack (EX Ruby & Sapphire in English), to the Korean-exclusive Start of an Adventure set in 2010, after which nine more similar Korean sets followed - and after that it mostly followed suit with Japanese for the BW/XY/SM/SwSh eras. The only other ex era Korean cards besides the ADV Expansion Pack that I’m aware of are the Treecko/Torchic/Mudkip Constructed Starter Decks, and a single 1/P Pikachu promo. Fun fact: all these Korean ex era cards have new Japanese backs, instead of their typical international backs. The Korean ADV Expansion Pack cards are also all in 1st edition, even though no unlimited edition version exists.
As for the question in your title: “Do Korean Gold Stars exist?”, nish already answered that.
Greetz,
Quuador