Timed Durations: Many durations are measured in rounds, minutes, hours, or other increments. When the time is up, the magic goes away and the spell ends. If a spell's duration is variable, the duration is rolled secretly so the caster doesn't know how long the spell will last.
The GM rolls, when it expires he proceeds with the next initiative order in the original round.
Ill Omen does stack with a follow up spell that has Persistent Spell on it. You roll twice, then roll again if your lowest roll made the save.
Ill Omen makes you roll twice and take the lower number before you know if a single roll would have succeeded or failed. Persistent Spell makes you roll again if you made the save.
Example: Target is under the effect of Ill Omen. He's hit with Persistent Charm Person, needs a 7 on his roll to save.
Step one: Roll twice, take lowest, as required by Ill Omen. Was the lower resultt lower than 7? Keep that, you fail the save, don't worry about the Persistent Spell effect. Was the lowest result 7 or higher? You would have saved, so Persistent Spell triggers, roll again and take that result.
Of course it's faster and the same to roll three times and take the lowest, but the above steps show why that is and why they are different effects that do stack.