Ah yes, good point. In that case, it would explain why she apparently was not pursuing Lynx. Since Lucca would have died fighting Lavos in 1999 AD, Kid was probably raised by someone else in that case.
She's not pursuing Lynx in Home World because Home World Lynx died with the Dragoons in the Dead Sea. So no word of Lucca's killer on El Nido, she has no reason to go there.
No, Kid was raised by Lucca in both worlds.
But I am still not convinced Kid MUST be in Home World. It is possible that Lucca dying in 1999 AD would in turn cause changes prior to 1010 AD that were dependent on Lavos being defeated. A change in the future that affected the past, prior to the split. Otherwise there would be a paradox that could not be explained. If Lucca finds Kid in 1004 AD because it was before 1010 AD, but died against Lavos in 1999 AD before said event takes place...well you see the problem. So the only answer is a butterfly effect where some events prior to the split change due to alternate outcomes that took place in the future, after the split. The way I see it, this eliminates Lucca from 1004 AD because she never visits that time before defeating Lavos. So if this would work for her, there is no reason Kid would be any different.
Are you sure she died in 1999AD? Besides, by the rules of CT time travel, the Lucca that finds Kid in 1004AD won't be affected if suddenly something meddles with a past self of hers. More so when that past self is in the future. No matter what happens to Crono and company now, they will still return to 1000AD, the version of them that succeeded and made the time travel back. Read up on Time Traveler's Immunity that the Compendium came up with, it explains it quite well why Lucca would still exist in 1004AD so long there's no further time meddling before 1004AD.
Also, you're using the wrong kind of linear thought. Lucca finding Kid is after she helps defeat Lavos, yes, but that's only in her personal history. For the time line, it happens way before. Therefore, the time line can be altered at any point after 1004AD and there would still be a Lucca, victorious against Lavos, that emerged in 1000AD and finds Kids years later, because it's not 1000AD itself that's being meddled. Only 1999AD.
In fact, the most accepted reason that the bad future returns in Home World is because the split essentially created a new timeline from scratch. Crono and company's time traveling is still bound to Another World. Which means it's more likely there simply was no Crono and company to emerge in 1999AD, since said time travel TO 1999AD wasn't copied.
Lavos does not die in Home World, so there is no fusion with Schala to create the Time Devourer in this timeline. This was the event that prompted her to make a daughter clone to begin with, so there is no reason to expect she did. It makes the most sense that due to an alternate outcome from the future, Lucca and Kid are eliminated from the present (in their own way). Furthermore I do not see TTI playing a role because this is not just a change in history, but an entirely new timeline. Logic dictates that in this case, this would just be another alteration based on possible outcomes that was caused by a butterfly effect due to a crucial change actually taking place in the future, but had direct influence on another event that would have taken place in the past.
Once again, wrong kind of linearity. Lavos also has the same Time Traveler's Immunity Crono and company have. This is because Lavos wasn't physically within the planet, he was in a sort of pocket dimension. His emergence in 1999AD was him leaving the pocket dimension, which is also a kind of time/dimension travel. Thus, his appearance in 1999AD is protected, as its a past version of him. Said version will still exist, even if his present self is now in the DBT, merged with Schala. Keep in mind that while we call them worlds, Home and Another are more accurate to two intertwined timeline in the same world/dimension. Another should've been discarded to the DBT when Home got created, but that didn't happen. It's an anomaly, but it's still one dimension. So, Time Devourer exists for both. And again, by Time Traveler's Immunity, Kid's entry to 1004AD is protected, no matter what happens to the time line. Maybe if she had been born after the future split, but that didn't happen.
To clarify with an example of the inverse, Dalton would still raise the Porre military to attack Guardia in Home World. The event that caused him to do so took place before Crono and gang had their final battle with Lavos. He would have had no way of knowing they were dead, so he would have acted out the same way he did before the split. The reason Kid and Lucca are affected is because the event that prompts Kid being born in 1004 AD and Lucca finding her takes place in the distant future, long after the split.
Again, wrong kind of linearity. There will be still a Crono, Marle, and Lucca for Dalton to plan his revenge on. So long nothing happens to them before Dalton's own arrival to the timeline, which might've been still in 1000AD, then nothing changes there. And again, Lucca and Kid will still be there, no matter what happens to Lucca's past self in 1999AD, or to Schala.
As for references to Radical Dreamers, they could have carried on without her. No where did it clarify that Kid was the leader of said gang, nor that she founded it. So the gang of theives being active on the mainland has no influence over Kid's existence in Home World. But even if for some reason she was born, she would have had to have been raised by someone other than Lucca. I suppose there is that quote from Kato that suggests she might be on the mainland, so it is possible. However that could just be him being ambiguous, which he did A LOT with Chrono Cross. I just do not see any in world evidence to back it up and considering how Cross ended, there would never be a need to go back and clarify such a comment in a potential sequel. He could have said pretty much anything and it would have had no impact on future plans he may have had at the time, nor the overall plot in Cross.
No, it was indeed founded by Kid and is the leader. This is explained better in Radical Dreamers (the game), where the group is just her, Magil/Magus, and that world's Serge. Again, there is going to exist a Home World version of Kid since she was inserted into the timeline pre-split, her arrival is an assured thing no matter what happens to the events that led to her creation, and said Home World version just simply doesn't have a reason to go El Nido, hence her non-factor in Cross's story. And the whole Dreamers thing comes form Lucca. She did a similar thing with Nikki which inspired to name his band the Magical Dreamers. It's not a coincidence. Both Kid and Nikki named their groups after Lucca's talk of dreams.
Either way fits into the theory nicely, but I strongly lean toward her not being present at all in Home World. There is no solid evidence that she is present, and the events that cause her to be born are a clear alteration to the timeline as far as I can tell. This is all of course, under the assumption that Crono, Marle, and Lucca all die fighting Lavos in 1999 AD in Home World, which is implied to be the case.
So to recap... no, she definitely exists in Home World. Her arrival to the timeline is a protected thing from any past alterations, and since it was pre-split then she was split with the worlds. Same for Lucca's, her arrival back to 1000AD from 1999AD after she beat Lavos is protected. So even if time changed in 1999AD so she dies instead won't affect present Lucca because that happened to a past version of Lucca. Again, it's more likely that the past versions of Crono and company that were meant to arrive to 1999AD to fight Lavos simply... didn't. And as established by CT, what happens to past versions of characters do not affect their present selves.
As for why would there still be a past version of Lavos to emerged in 1999AD? Well, the pocket dimension is its own thing. By it's very name, it's contained within the world, even if it's not physically present. So the pocket dimension gets copied, Lavos still comes out on schedule. Though this last part is just conjecture on my part.
If you want further proof of Kid existing in Home World, Ultimania has this:
https://www.chronocompendium.com/images/wiki/4/42/074-075.jpgNotice how Kid's Home World status is a dash. This denotes "Exists, but does not appear in the game", when you consider the other characters that have that dash. Characters who we know have to have world counterparts, but we simply don't know what they're up to in the present. It's certainly different from the X that denotes being dead (like the Another versions of Serge and Marge, and the Home version of Direa, for example) or the ? that likely denotes "Fate unknown", though we certainly learn about some of them, like the Home Dragoons. If Kid didn't exist in Home World at the present, she'd have the ?, or even X.