But they couldn't *know* that Elizabeth was the one they were looking for. They could consider her a possibility--she fits the description somewhat--and start out talking to her to see if she's suspicious. It was her decision to not try to talk her way out of it.

I may be misremembering, but didn't Oleg get the right to look at Nina's file in the last series?

He definitely got himself more security clearance using his connections at one point, and that gave him access to Nina's files. I can't remember if that was his specific stated reason for it--it might have been.