On the question of whether or not a Polish person would understand Russian, I can contribute some anecdotal evidence. My mother grew up speaking Polish at home, and studied Russian very briefly as an adult through an educational television program, but never took any formal classes in Russian. One time, she attended a business dinner with my father and some Russian guests of his company, and told me later that she was able to understand a good deal of what they said to one another in Russian. So if FS, who grew up speaking Polish, set her mind to learning enough Russian to understand conversations, and to interject the occasional comment in Russian, she had a head start as a native speaker of another Slavic language.
Just anecdotal evidence, as I said. Personally, Anna Anderson's poor English is one of the things that convinced me that she was not Anastasia, but I also don't find the evidence that she spoke fluent Russian very convincing.