Since I haven't watched the show in great detail yet, I can't say for sure who the king is. But considering the Bridgerton setting which draws from the Regency era, it's probable that the king is modeled after one of the Hanoverian kings. Given Queen Charlotte's presence, it might be George III as he was her husband in real history. However, the show may take some creative liberties with the character, so it could be a somewhat fictionalized version of him with added elements to fit the Bridgerton story.
The king's fate in the story might have involved some health issues or political complications. But you'll have to watch or read to get the full details.
I don't think King George is explicitly depicted as gay in the series. It could be a topic of debate among viewers, but there's no definite portrayal of him being gay within the narrative.
The role of the king in Queen Charlotte: A Bridgerton Story is taken on by Corey Mylchreest. His performance has been well-received by viewers for bringing depth and charm to the character.
It provides a backstory to the Bridgerton world. It shows the relationship between Queen Charlotte and King George which might have influenced the society in the main Bridgerton storyline.