You don't lose your native accent just because you live abroad.
Believe me, I know what I am talking about.
She is acting bizarrely or having you on.

Maybe after a lifetime away from a country you only spent your childhood in.
But not in under a year.
She's not even going to be a fluent speaker of Russian in that little time - it's a hard language.

As for her name, yes, the regular Christian names have an equivalent in all European languages, including Russian. Personally I prefer English people to use the anglicized version of my name, or an English nickname, because it just sounds ugly when they have a go at the real pronunciation, plus it just singles you out unnecessarily as foreign. She probably reached the same conclusion - also, the Russian nicknames are rather cute, I can see how she'd enjoy being called Katia or something, even if she is a "Kate".