A580 I think we need to know more about Eve's back story to answer your question. We know far more about Villanelle's. Perhaps in the second series? There was all that private research - not part of her job - she'd done into female assassins before Villanelle really came on to the radar, which got her the job with Carolyn.
They'd already met in the hospital bathroom and then there was the present of the perfect clothes and 'Sorry, baby', which may have given her the idea of a personal connection which would protect her when she got out of the car?
I think one of the reasons a second viewing works better is that it takes you some time to work out which genre you're dealing with. It's quite ground-breaking in that respect. Black comedy? Thriller spoof? But then all that changes with Bill.
It's quite subtle the way Villanelle's mother obsession is built up. When she's out walking with Sebastian, she turns her head when a passerby with the same length hair-style as Anna and Eve catches her eye. And there's the American tourist of a certain age. And the drawing of a faceless woman which upset her at the psychiatrist's. I think she was too young when her mother died to remember her face.
I loved the 'fruit cake in the freezer' detail. That was a dead give-away about Frank's 'mother'. No English woman would freeze fruitcake. Villanelle seems to have twigged that it was a bum note.
The awful child was genius. Hope we see more of her.