When Yu Yevon was finally defeated, his greatest summoning ceased to be - the Dream Zanarkand. Yevon's power gave strength to the Fayth, enabling the Aeons and the Dream World to exist.

Tidus emerged from the dream, just like the Aeons do when a summoner calls on them; he became as real as anyone else, though before he had been just a product of the endless dreaming of the Fayth, rendered into life by Yu Yevon's eternal summoning. At the end, the power of Yevon and the Fayth has gone; the dream is ceasing to exist. As Tidus leaps from the airship, he does indeed come to the Farplane - Auron and Braska are waiting for him, and Jecht too, which adds proof that these 'dreams' are real enough that their souls live on. Now, Tidus didn't reach the Farplane because of the Sending, but then the game hints that the sending isn't always necessary for a soul to reach the Farplane. After the credits, Tidus awakens in a vast sea, just like the Fayth promised he would. Shiva, I believe, says that this is their parting gift to him.
Shuyin isn't the real Tidus. The whole Yuna/Lenne Tidus/Shuyin thing is merely dramatic coincidence. It's similar to the coincidence between Laguana/Squall and Julia/Rinoa in FFVIII. They are two different sets of people who happen to be going through the same thing, and in FFX-2's case happen to look a lot alike. Shuyin was real 1000 years ago, but Tidus isn't the recreation of him, at least there's nothing that mentions that. He just merely looks like Shuyin, as Yuna just looks a lot like Lenne. It's nothing more than dramatic coincidence.
Here's something to consider... Lenne was a summoner, so are all the other Fayth; it's conceivable that the other summoners knew Lenne - and Shuyin. Tidus is a product of their dream, so maybe he has some of Shuyin's characteristics too? Kind of like he's the embodiment of fragments of the summoners' memories, a little bit of the Fayths' former lives coming into the dream. It makes sense... and anyway, Yuna doesn't really look much like Lenne at all, just when she's in the Songstress form. Shuyin and Tidus have far more in common.
The fayth were not killed, the aeons were killed. The fayth merely stopped dreaming. A fayth is a person who gives his/her soul to Yevon so that a summoner can use the fayth to call create an aeon.
The Fayth were originally Summoners of Zanarkand; following Zanarkand's defeat by Bevelle, the summoners agreed to have their souls trapped in statues so they could preserve the memory, the dream of what used to be. The Fayth are those imprisoned souls. Technically, they are dead; Auron says as much in the game. The Aeons are manifestations of the Fayth, the power they used to have, come to life again to aid in the fight against Sin. "We Aeons", as the Fayth of Bahamut said.