But there's another that we're missing. A bundle of white flowers are left on the tomb. Today I noticed that Gwynevere was leaning on a pillow with a white flower design, and that made sense to me.
There is a fourth descendant of Gwyn, after all; Priscilla. And what would she have to offer but her doll?
Take a moment to let that sink in while reading this: The upper room of Havel's gear in anor londo is covered in water. On the ring of the firstborn, there's a line about how his foolishness led to the destruction of the annals. In that room there are what looks to be scrolls. More likely than not, the firstborn's actions there led to the destruction of that room filled with records. Why did he do this? He didn't. His actions led to the destruction of the annals but he did not destroy them. They were destroyed by someone else and he was blamed for it. But back to the doll.
The reason the doll is missing is because the firstborn took the doll. The reason his corpse is outside of the tomb is because he was killed while leaving. Most likely the firstborn lost his life while escaping from Gwyndolin after having been stripped of his deific status, taking the doll along with him. When he died, he was sure through his actions that he would be marked as an undead, so he would need that doll that his lonely niece held so tightly before she found friends in the painted world.
I believe the firstborn is Solaire of Astora. Note two things in Solaire's dialogue:
1) He mentions that his journey is lonely.
2) He calls himself a fool.
This further solidifies my point; Solaire went to the same asylum as you. Note that the crestfallen warrior, I believe, says that someone went to the asylum curled up in a ball. This could be Oscar, or it could've been Solaire, to put the doll back in the cell he had come from. Now why would he do that?
I believe he did that because he found you. You are now his friend. He's not lonely any more and therefore has no use for the doll. However, he does not wish to trample upon his father's grave yet again, so he leaves it somewhere more symbolic; the place that he clutched it so tightly, hoping for the day he'd be free to search for his own sun and his close friend.
For the fool bit, well, this part's more of a stretch, but I think it goes back to the point about his foolish actions leading to the destruction of the annals.
Conspiracy theory here: I believe he accidentally discovered Havel's hidey-spot and found the dark club there. He notified Gwyn, but either in the ensuing struggle or something else in that process, the records were soaked and thus destroyed. I believe him wanting to find his own sun is related to this event. Gwyn hid that all this happened, so it's very possible Gwyn himself destroyed the records specifically with water to hide the flame marks from the battle that took place between him and his friend Havel. Solaire saw this as corruption and like Havel began to question Gwyn. He wanted to find his own sun rather than inherit the corrupted sun of his father. This is why his powers and godly status were taken from him. Yet he was still allowed to stay in Anor Londo as he was still the son of his father.
Now this is all speculative, but I think it's a possibility in such a situation given what each character is like. Solaire is an explorer, Gwyn is corrupt, Havel was trying to pick a fight with Gwyn over his rule, Priscilla was a lonely person until she made it to the painted world where she described its inhabitants as kind, and Gwyndolin just plain old hates anyone who sees him in a dress.