It might help if you think of Dream Zanarkand as a giant Aeon, the product of the collective unconscious of counless Fayth entombed on Mt. Gagagzet. Like any Aeon, DZ exists in a solid state of being as long as the Summoner continues to summon (which Yu-Yevon has been doing for 1000 years).
Unlike a regular summonable Fayth, where a single corresponing Aeon appears, the makeup of DZ has shifted and evolved over the millenium as people are born and die. So each individual in DZ is an expression of the collective dreams of the Fayth, rather than their individual dreams.
There's nothing to suggest Tidus has Shuyin's soul, although no doubt many Fayth knew Shuyin personally and his appearance was the model for Tidus. However, there's nothing to suggest that it was a deliberate act on the part of the Fayth.
I think Tidus existed even after the Fayth woke up, just in a non-coporeal state.