The rules are to finish what triggers the final cutscene. In the ps1 and pc versions you have to complete the randall race while, in the ps2 version, you can quit out of the randall race to trigger the cutscene. This means that the time should stop after the last medal is collected. I don't know when the rule was changed but I also don't run ps2 version.

Emulator is up to the runner. I personally use ePSXe so that is definitely not a problem