The dimensional split occurred in 1004 A.D. Lavos goes undefeated simply because there weren't two dimensions at all when Crono and the gang originally took him out. While historically, time traveling that predates the split might be preserved somehow, it isn't in the future, since Crono is still departing from Another World to the future of Another World; he cannot cross dimensions with his time traveling. This is the background for the Armageddon Branch Theory. Suffice it to say if you subscribe to that theory or not, Crono is defeated this time around, or prevented from finishing his mission in Home World. It is unique to Home, and Home is the anomaly.
The Time Devourer hasn't matured, and if he had, the game would have ended, as all space time continua would be immediately consumed. He's out of this really.
FATE keeps the Dead Sea around because the Home World version of the Frozen Flame is still there. To explain this and the other stuff, know that the entire Sea of Eden exists ten thousand years in the future, and entering it is like using a Gate. When the dimensions split, the future became ruin in Home World, and instead of a nice little Chronopolis in it, we have destruction. It exists, it just hasn't "happened yet" and won't until 1999 A.D., most likely. The talk that it is a future that isn't supposed to exist, I believe, it symbolical; Crono was supposed to save the future, but in Home World his efforts are nullified and that future comes back into existence.