The Traveler is extremely important. They are the main protagonist, on a quest to find their sibling. Then there's Paimon, the Traveler's sidekick, who provides comic relief and also helps in explaining things to the player. Zhongli is also crucial. As the former Geo Archon, his knowledge of Teyvat's history and his influence on Liyue's development are significant parts of the story.
Sorry, I can't pin down who the blue-haired character is in that bedtime story. It could be one of several characters. Try looking through in-game dialogues or character bios for a clear identification.
As of now, the Genshin manga is not finished. The story is still evolving and there's more to come for fans to enjoy. The creators are taking their time to develop a rich and engaging narrative.
There's no definite answer yet. It might continue if there's enough demand and the creative team has more story to tell. But it could also stop if they feel they've completed the story arc.
There is a Genshin Impact manga available. It brings the game's world to life on the page with beautiful art and engaging storylines. It's a great companion to the game.