This is something that's been bugging me for some time, and after thinking about it, I've come up with the following. Peruse at thine leisure; respond if you wish. It's, really, really long, and it took forever to type, so I ain't editing this just now.
On the subject of Kid and Schala...
I don't think, given the dialogue of Lucca, Kid, and Belthasar, as well as the experiences of Kid and Schala, that the two would merge into one being. Several points to support this:
1. Kid is vehemently against the idea that she IS Schala, and Lucca says that Schala would have wanted it that way. This suggests that Kid is, in fact, her own being, independent of Schala, and that Schala would respect this. This is a bit tenuous, since we don't hear Schala say this herself; however, given that Kid and Schala are similar, Kid's opinions would be somewhat akin to Schala's, no? Either way, I think Schala would take a benevolent view of Kid's identity, and allow her to be her own person, if possible.
2. I don't think that Schala, having experienced aeons of suffering after fusing with the various entities in the Darkness Beyond Time, would forcibly merge herself with Kid, if she could prevent it. And we're told that Kid is a copy of Schala, not a part of her, so it may not have been necessary for them to fuse at all.
3. Belthesar is very careful in his dialogue when he appears in Terra Tower. He states that it is Kid who travelled back in time to save Serge at Opassa beach, creating the dimensional split, and that it was Kid who called Serge across the dimensions. He also states that Kid traveled from ten years in the future to accomplish both aims, from the year 1127 A.D.
Given this, I would say that the entity known as "Kid" must still exist in the future, and is capable of autonomous movement.
4. Some of the strongest evidence against the idea of Kid and Schala existing as independent beings appears in the credits of the "good" ending: in this ending, we see someone with a more feminine appearance than Kid's at Opassa beach, assumably to summon Serge into the other dimension. Also, the journal is signed "Schala 'Kid' Zeal".
But I don't think these two things mean that Kid and Schala have become one. Rather, I think they mean the exact opposite!
Remember, we're dealing with events ten years in the future. In 1117 A.D., about the time that Chrono Cross takes place, Kid is only 16, and she's a pretty headstrong youth. It's fair to assume that by 1127 A.D., Kid has grown and come into her own; many people have similar experiences as they grow up. This would explain why she would appear so different without necessitating her merging with Schala.
As for the journal...it is wholly possible that Kid, having no real name besides "Kid", would eventually come to terms with her identity as (at worst) Schala's clone, or as her daughter, and take on Schala's name. But this would still allow her to retain her own name. Lucca alludes to this in her letter to Kid. Also, if they didn't merge, Schala would have no reason to take on Kid's name, since she was never called by it.
5. Further, we see Kid traveling in her thief's costume some time after the events of the game. To me, this suggests that someone with Kid's persona is traveling with some intent, which goes along with the idea that Kid finds Serge after the events of the game.
I don't know whether she is leaving El Nido or returning to it, or whether she is just traveling through it (since a large vessel is needed to traverse the currents to leave the archipelago at certain times of the year). Simply put, we don't know where Kid would have been left after the final battle. Everyone else would have returned to where Serge found them, with some variances, given that they would then be in the Ideal Timeline.
However, this raises a question: where is Kid? She never appears in Home World; even when we visit Guldove in Home World, her place in the clinic has been taken by Orlha's sister, Tia. We are never told what became of Kid is Home World; for all we know, she was killed by Lynx and Harle. That, or she's on the mainland.
Given this, it's safe to assume that Kid returned to either the mainland or El Nido, and began to search for Serge. After all, there's no reason for her to forget everything; in fact, it would create a paradox, since it's a future Kid who is supposed to create the dimensions and save Serge.
6. One thing that drives me crazy: if Schala was killed, or sent to the another dimension, how on earth does Nadia receive the pendant? Are we ever given the identity of Nadia's ancestor from the Dark Ages? If not, I assume it's Schala, since that's the only one who could have passed on the pendant.
But that just brings me back to the point: how did the pendant, which Schala is supposed to have given to Kid, come into the hands of Nadia in the modern era?
I doubt that Kid traveled back to give the pendant to Nadia's ancestors, whoever they were.
(On an aside, I should note that Kid would need to enlist the help of someone who knows the mechanics of time travel in order to save Serge. She could have called on any of the Crono Trigger heroes who live in the present, but this would necessitate one of them being alive, having survived the Fall of Guardia and, in the case of Lucca, having escaped Lynx.
Thus, it would be safe to assume that one, or all, of the main cast survived in the Ideal Timeline.
I suppose it could have been Belthasar, but we don't know what became of him.
But this all depends on whether Kid still possesses the Time Egg she assumably took from Lucca's house. In Chrono Trigger, the Time Egg can only be used once; in Chrono Cross, it takes the powers of the Masamune, in tandem with Doreen, to be able to travel with the egg.
Was the egg used up in saving Kid? Did the egg used to enter the Darkness Beyond Time shatter also? Even if she had the egg, Kid would have additional power in order to save Serge. But I digress. Really, really, digress.)
This brings me back to a line from the "good ending," where someone (Schala or Kid) tells the party that time will be united again. If so, then the powers that merge time and space would restore each person to their original periods, just as the party returned everyone to their various time periods at the end of Chrono Trigger. Logically, then, Schala would return to the Dark Ages, restoring the time line of the pendant.
Kid's place would be unsure; however, since her origin is a place that exists outside of time (the Darkness Beyond the Time), I think it's safe to assume that she would return to the Present, where she first appeared, or would be given a choice between eras. And she's never been to the Dark Ages, so why would she go there?
I think this is what really clinches the idea of two separate entities; how, otherwise, can you explain the idea that Schala's pendant - which she took to the Darkness Beyond Time, and later bestowed on Kid - end up taking part in the plot in Chrono Trigger? If Schala is never saved, as we are left to believe in Chrono Trigger, then there's no way that the whole adventure could have taken place!
6. Building on the previous point, if Schala and Kid did part, and Schala returned to the Dark Ages, it would make sense for her to search for Janus, as has been speculated. It would also explain the bit of dialogue from the "good ending" where the unidentified speaker says that the two people might not know each other. (It could also apply to Serge and Kid, I suppose, what with his seeming amnesia). If Janus has already lost his memory, he wouldn't know his sister; and if Schala has been trapped for aeons without seeing Janus, keeping in mind that he is capable of transforming himself and other beings (Glenn, for instance), it would make sense that she would be a bit unsure of his identity.
Logically, if Kid and Schala are two independent beings at the end of the game, then the dialogue that concerns searching for someone can be applied to both, suited to their own circumstances. They both have a person they care deeply for; in Schala's case, it's Janus, and in Kid's case, we can assume it's Serge.
On another aside, I don't think Janus would be satisfied to only find Kid, since Kid has none of the memories of his sister. I don't know if he would continue to search if he was to find Kid, but I don't think he would be content until he found Schala.
If you've read all of this, my hat is off to you.