My understanding of it currently is:

1. Her skill tells her when and where her students are going to die.
2. Since it doesn't tell her their current location, the only reliable way to find them is to be around when they're scheduled to die.
3. She tells her father, and he has them scooped up before they die.
4. This is the reason there's a whole group of uncollected students at the start of the series. She didn't have them taken because they weren't in danger of dying.
5. Oka 100% didn't "happen" upon Schlain and Hugo earlier on. Her power told her "Schlain gets murdered here." and she was expressly there to prevent his death.
6. This is just my assumption, but I assume when she prevents a death, her list gets updated. Since she's able to check it again later.