1. They had a plan, however briefly, but then Je-yi decided to switch to "Plan B". Plan B didn't involve Seul-gi (and was probably Je-yi's original plan before Seul-gi got involved). However, Seul-gi knew that Je-yi's plan would end up incriminating her, so because she didn't want Je-yi to ruin her life trying to take down her father, she took action on her own. So she ultimately swallowed the pills, so nobody could get the exam questions. This obviously backfired immediately. So, Je-yi, who never actually wanted to involve Seul-gi like this in the first place, had to improvise, which led to the surgery.
2. She wanted to escape the toxic environment she was in, and she thought this was the only way to do so. Obviously using the collected evidence would also damage her father, but the prime motivation was escape. She probably knew full well her dad would never remain in prison (since she knew he didn't kill Seul-gi's dad, so there'd be no evidence of that), so escaping like this was the only solution she could think of.
3. Je-yi is creative, so she'll find a way. She had a while to prepare for this, so she probably had a few plans (and money) in place to actually buy herself a new identity, and on how to lay low. And, well, nobody's going to bother looking for you if they think you're dead.