The problem with the first bold part is that Marle disappears after Crono travels, but before Lucca does. Lucca doesn't arrive in 600 AD until after Crono is already in jail. Therefore, whatever caused Marle to disappear had to happen after Crono traveled, but before Lucca.
No offense intended, but you need to play CT again. Crono doesn't get put in jail in 600AD; that happens in 1000AD, after all three safely return. Lucca appears as soon as Crono gets downstairs from watching Marle disappear in a shower of sparks; she had to already be on her way to the castle for that kind of timing.
I disagree with the second bold part entirely. I don't think Frog's decisions can influence the future at all. Once Marle travels, the future is set in stone: in the original timeline, Frog made certain decisions. The only thing that changes is Marle being present. Her presence either will either affect Frog's decisions, or they won't. And the effect they will or won't have should happen instantaneously, changing the future immediately. I don't think Frog really has any say in the matter. So, if Marle's disappearance was based on Frog's new future decisions, she should've disappeared on the spot as well. Also, that would still attribute Marle's disappearance to the Grandfather Paradox, which we don't want to do.
Well, that is why I said personally, but allow me to explain it a bit more to illustrate how it
could be possible. Frog makes a decision (I would say that the decision was to rescue Leene and elope with her, being disillusioned by the King's inability to tell the difference between two distinct individuals), so time would change instantly, as you pointed out. But, he has to make that decision first, doesn't he? So first he has to hear that Leene has been saved, then he has to travel to the castle to find out that no she hasn't been saved. He might make the decision right then and there, or he might have waited some time (perhaps even waiting until he found out that Lucca was looking for the princess). Thus, such a decision could have come any time between when Marle was found and Frog met Crono and Co. (my supposition then being that he made such a decision and then, immediately, Marle disappeared).
Frog deciding to elope -> Leene has no kids (or has frog kids) -> no Marle is still a version of the Grandfather Paradox, even though Leene is not killed. The point is that Marle should not disappear for that reason, since her presence in 600 AD is protected by TTI.
But that is specifically it. By defining when the element that causes Marle to disappear occurs, we can determine if Crono and Lucca would be subject to TTI or not. As for Marle's disappearance, that will always be a bit of a conundrum. Her time travel was a bit non-standard; is TTI explicitly the result of time travel, or do the means of Time Travel bestow it; does TTI only protect a person's appearance in a time period but not subsequent deletions; etc.
There are two different problems and, by supposing that Frog might have been behind Marle’s disappearance, I am only really addressing one.
The first problem is that if Marle’s presence in the past changed the timeline, Crono should have never been able to travel to 600AD in the first place (and likewise, Lucca shouldn’t have been able to). The only supposition I can think of that would preserve the story is to assume that the event that caused Marle to disappear happened after Lucca had traveled to 600AD. That thereby implies that Lucca’s presence in the past somehow effected the timeline (possibly in the “straw that broke the camel’s back” sort of way) which is what caused Marle to disappear. Such a supposition avoids the first problem of the paradox.
The second problem with the paradox is, even if Crono and Lucca were able to travel to 600AD, any change to the time line would not have affected Marle because of TTI. The Entity might have been responsible, but such an answer is unsatisfying and unsupportable. Unfortunately it is a question of 4th dimensional physics, which we know very little about (in the game or otherwise). I don’t really have a good answer to this one problem (yet).
My thought on it is that the Entity caused Marle to disappear. Marle's disappearance is what gets Crono thrown in jail. Crono's conviction is what gets the Kingdom of Guardia (or the Chancellor/Yakra) mad at him. Being mad at Crono is what gets them chased into the forest. Being chased into the forest is how they find the gate to 2300 AD. Taking the gate to 2300 AD is how they decide to save the world.
So, the planet made Marle disappear, to lead Crono and co. on a quest to save the world.
That would work... if the Entity intended to have Crono & Co save the world. The question is, then, if the Entity so intended. The evidence is not very supportive.
First, the only evidence we have of the Entity's intentions is in the scene in Fiona's Forest. There it is supposed that the party's adventures have been the result of some entity reliving painful memories before it dies (in the specific case, memories of Lavos). There is no mention of the possibility of that entity bringing Crono & Co into the story in order to change those memories (though there is mention that memories can be regrets accompanied by "if only..." thoughts); the closest we get is an implication that the party are meant to be passive observers, sitting with the entity in its last moments of life.
The full implications of the Entity being behind Marle's disappearance are a bit larger than you portrayed. Yes, it would start the adventure, but it would also have taken the decision out of Crono's hands. If "The Entity Did It," Crono & Co are no longer free agents capable of changing the future of their own volitions. They go from being heroes to pawns. While such might be possible, it goes against the theme of the game and, as the game is a story first and foremost, I must reject it.
Side note, Crono being with Marle, who had disappeared, is what was responsible for Crono's jail time, not the fact that Marle ceased to exist in 600AD. There is no indication that her father was ever aware of those events.