Traverse Town is a world made from the remains of destroyed worlds. Everything in it is just an amalgamation of the ruined portions of worlds consumed by the heartless. It's not that Sora intentionally used any of the dark corridors or anything like that. His heart was simply strong enough that when his world was completely consumed he appeared in Traverse Town.

This is also presumably why Traverse Town does not make appearances after Kingdom Hearts. Once the worlds are restored at the end of Kingdom Hearts, everything that made up Traverse Town returned to its original world (as we see Kairi is returned to Destiny Islands when it is restored).

Regarding Kairi, we are first shown her body in an image by Maleficent. From what Nomura has suggested in interviews, it's probably a safe assumption that when Kairi's heart separated from her body, the body was lost in the Realm of Darkness. Another safe assumption would then be that Maleficent recovered the body from the Realm of Darkness. As for how her body ended up on Captain Hook's ship, it's probably a safe bet that Riku was keeping the body with him on his way to Hallow Bastion.