1. She was created as a sort of agent for the Dragon God topside, doubleplaying Lynx and later Serge. Unfortunately, she fell in love with Serge, making her job pretty tough and giving her some grief. Her final purpose was to get Serge to defeat FATE and let the Dragon God retrieve the Flame, which she accomplished somewhat indirectly.
2. The Frozen Flame is just intrinsically corrupting; it's a magnificent power. What turned Wazuki into Lynx was the manipulation of his going crazy by FATE, which came to use Wazuki as an agent against Serge, the Arbiter. When Serge made contact with the Flame as a baby, it made him the Arbiter of the Flame. Now, only the Arbiter of the Frozen Flame can access its power; this is a major problem for Chronopolis, who uses the Flame as a power source. The mechanical thing determining who can use it and who can't is a little machine called the Prometheus Circuit. Before Chrono Cross, when Chronopolis had been built and Belthasar was just learning about Schala being bound to the TD, Robo was still around. Robo volunteered to be part of the circuit and Belthasar's plans; FATE had no idea he was in the machine until that night, when he locked out FATE from the Flame because Serge was the new Arbiter.
So, Lynx is the agent of FATE trying to get the hell rid of Serge so the lock will disengage. He already tried kidnapping Lucca to have her dismantle it herself; she apparently resisted and died. He probably torched the orphanage out of sadism. However, after learning of the Dragon Tear and getting Hell's Cook, Orcha to steal it, he was able to simply copy Serge's genetics, which qualified him as the Arbiter anyway.
3. The Dead Sea's only been there since 1010 A.D., since the dimensions split. The entire Sea of Eden sort of exists in a future type state; this is explained further in the Salt for the Dead Sea article by GrayLensman. Now, when the dimensions split, the intrinsic differences of each determine that Home World is going to end when Lavos blows up in the future again, effectively undoing Crono's earlier actions (why this occurs is a popular mystery in the series, and is also addressed in the article). So, when the dimensions split, Home World's Sea of Eden, which exists in a future state you recall, changed violently to reflect the future of Home World -- destruction.
I know that's really "What? Why make it so complicated?", but the Dead Sea is singularly the most enigmatic aspect of the Chrono series. And I mean, enigmatic as in "what the hell," not "what happened to Magus?".
