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 Bridgerton series often features a fictional Regency-era society with its own take on royalty. In the story, the King and Queen play important roles in setting the social tone. The Queen, for example, is highly involved in the marriage market among the ton. Their presence adds to the grandeur and the complex social hierarchies that the characters navigate. They are symbols of power and propriety in this fictional world.
The Queen is very observant. She seems to know everything about the social scene. The King, he has an air of authority. He is more in the background but his presence is still felt. The Queen is very much into the social events and matchmaking, which shows her interest in maintaining the social order.
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.
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.