'A necktie slung inside of it' The key here is to remember that what we call English is not the Queen's English that the book was written in. Jessie Burton is English, which can affect her word choice. In this context, the necktie slung inside of it means a women's scarf tied 'round her neck, and the ends tucked into her shirt, probably like the picture at the link below.
https://www.pinterest.com/pin/70861394106902948/
Your second question, what is meant 'raffing around London' is most likely to refer to how the British 'riff raff' acted, and has the connotation of partying around London irresponsibly, and acting beneath her class. (For some reason, the British are quite fond of their class distinctions.)
Your third question about 'A cat amongst the Blitz' does indeed refer to the German bombing, and it most likely is defining the young lady's character as someone who is somewhat reckless (like a cat), and yet somehow avoids the bombs (either of the Germans, or of life), much like cats are often able to do.