I don't know if he has *death* flags, but I do think that his Fate within Teyvat is to be a sad ending considering he even jokes about him and Diluc being "doomed together" However, Fate has been thwarted before and I think it is possible it will be thwarted again for Kaeya's (K) sake based on past plot points in the game.
Durin is a good example of this. As is Dvalin, since we now know that Varka leaving Mond allowed the Traveler to be the blonde hero to save the day which resulted in Dvalin being cured of his corruption.
I think K has a "doomed narrative" by Fate in the world of Teyvat based on his hangout quest as well as his comment about him &Diluc being "doomed together" to name a couple of examples K also mentions Fate a lot randomly in the game. I believe this is foreshadowing.
However, I think he too will be spared his "doomed Fate" because of the Traveler & their abilities as a Descender to manipulate Fate somewhat. It seems as though they have the ability to save all their loved ones who have visions from a bad fate.
I am not sure that Hoyo will actually kill someone off within the plot of Genshin at this point. It seems that they conveniently resurrect or spare their playable characters (not that I am complaining I would like a happily ever after to this story as corny as it might be)
I think Kaeya's hangout featuring the play scene only backs this theory up. Kaeya is planning on defying his Fate and doesn't plan on bowing down to whatever cruel fate Fate is trying to bind him to.
To answer your second question: yes, I believe he will be relevant in the plot later on. I specifically believe he will become a large plot point for the bridge from Teyvat to the Khaenri'ahn arc
Kaeya comes up for no reason all the time through the game. It's like Hoyo is reminding us he exists despite a situation having nothing to do with him. An example of this is his mention in the Chasm quest (which was a huge lore drop about the cursed Khaenri'ahians)
Or how he met Dainsleif and talked about his surname two years before Dains met any other playable character. Not to mention the Alberich clan is the very one who founded the Abyss Order that the Traveler's twin is now prince/princess over
The Alberich clan seem to be important to the plot. Especially considering Calibert Alberich proved to be a huge plot point to the story in regards to the creation of the Loom of Fate which was crafted for the benefit and welfare of Khaenri'ah's Ley Lines.
So I think that it is safe to say that Kaeya will have future relevance to the greater plot of Genshin. I honestly anticipate that he will get his full story and next year at this time at the end of the Snezhnayan arc as we look towards the Khaenr'iah arc.
