"Hearts" in the KH Universe practically cannot be "killed" or destroyed so that there is nothing left of it since they consist mostly of memories which are confirmed to be immortal in Chain of Memories.
They can be damaged and injured to the point that they cannot function anymore, but they are never completely "gone" in the sense of that there's nothing left of it.
That whatever is left of Ansem disappears from Riku's body is a fact as he returns to normal and cannot open dark corridors anymore. The core question is what happened to all the parts (like Ansem's mind, as Nomura stated that Xehanort, Ansem and Xemnas each have a mind of their own) that weren't needed to recomplete Xehanort?
In the original japanese, they use the word "kokoro" for "Heart".
"Kokoro" however has also additional meanings in Japanese which are:
- essence of the individual/sense of self.
Maybe that helps a bit to understand what Hearts in their entirety in KH actually are.
As for Young Xehanort, he had always a desire to see the outside world, but the stuff Ansem SoD told him and his time travel (which he all forgets once he returns to his own time, therefore "unknowingly") deepen and intensify that desire and as YX says, it is "etched" into his heart and so he follows "his heart" to do what is already engraved there.
Allow me to understand this...
Apprentice Xehanort was disobeying Ansem the Wise by making experiments on humans and somehow eventually starting the heartless outbreak. Eventually, he intentionally used his Keyblade to split his heart from his body. His memories and mind were shared between both his Heartless ('Ansem') and Nobody (Xemnas). Because he intentionally split his heart from his body, he had full control over his heart, but did not become a real heartless like Sora did, because the KoPH brings out the darkness in whoever it is used on, which is seemingly not true in the case of Xehanort.
Some time before splitting his body into a heartless and nobody, Apprentice Xehanort sent Kairi out to sea, as she was believed to be a princess of heart, and he believed that her being this would make her go to the future Keyblade wielder. After his theory proved true, this is when he split himself 'in two'.
As I was saying, the 'robed figure' (Xehanort's walking heart) went to the Destiny Islands, which is where Kairi landed (how he knew, I am not sure). After a long time of waiting, the robed figure confronted Sora, saying that the world had been connected, meaning that the heart of the world was falling prey to the heartless.
Much later, Xehanort's walking heart possessed Riku, being only a heart and having the capability to do this. Riku's own heart was forced out of his body, and sent to the realm of darkness. 'Ansem SoD' was eventually destroyed by the light that came from his Kingdom Hearts, seemingly destroying his heart and making 'his' body go back to Riku, who became fully resurrected in the realm of darkness. Ansem SoD's Kingdom Hearts was made from the hearts of worlds, and this Kingdom Hearts soon became 'dismantled'.
Through some means, Ansem SoD planted a fragment of his heart inside Riku's body while he was using it. This heart fragment could somehow show its physical form in CoM without using Riku's body. At the end of CoM, Riku seemingly defeated the Ansem SoD of this fragment, but other than being 'destroyed', it was just weakened. This heart fragment continued to have an effect on Riku, eventually overpowering him in Days and forcing him to use the dark power of this heart fragment to finally defeat Roxas.
Near the end of KHII, this heart fragment was somehow 'pushed' out of Riku's body because of the explosion of Ansem the Wise's machine, and this is when it became the 'robed figure' again, travelled back in time to pass on his ability to time travel to Young Xehanort, and told Young Xehanort that he needed to pick up other incarnations of Xehanort and (for some reason) also told him about all of the events that happened from before Kingdom Hearts to near the end of Kingdom Hearts II (stupid KHwiki).
Because Young Xehanort's mission at collecting the other incarnations was not finished by Re: Coded, it obviously took some time.
There are two holes in what I have said, though...
1. If what I said is true, then this means that Xehanort's wandering heart is still 'alive'.
2. The robed figure could not have been at Destiny Islands at the beginning of DDD if he was never defeated and if he did come back after being fully expelled from Riku's body; this would be because he cannot be in that past and the present at the same time.
I really want to be corrected here... I still do not see how the bold part works...