Timing starts when you select "Yes" after entering a file name. Timing for full-game races ends on the circle fade-out before credits.
The game needs to be booted via normal/vanilla means. This means that all homebrew apps and/or special loaders are banned from being used, even if the official game disc is being booted through them. This also means that emulators, MIOS modifications, and any of the Action Replay discs are also banned.
The specific rules for commonly ran categories can be found on their speedrun.com pages.
TTYD speedrun.com page
TTYD category extensions speedrun.com page