Question: Anya believes she has family in Paris, according to her necklace. However, if she had never met Dimitri and Vladimir and gotten there herself, how would she have been able to find her "family" if she did not remember their names, her real name, and last name?
Answer: Anya was desperate to locate her family and may have been willing to search for additional leads in Paris. Thousands of Russian refugees had settled in Paris after escaping the revolution, and they could have helped her, someone might have recognized her, and so on. Also, Anya's memory was slowly returning.
Question: In both Anastasia's ballroom daydream and nightmare scene, why is it her sisters are the only ones who age and not her brother?
Answer: Dreams are not reality and are subject to an individual's interpretation. Anastasia's brother, the youngest sibling, was sickly and physically weak. (In real life, Alexei suffered from hemophilia.) Anastasia apparently always sees him as an innocent, vulnerable child needing to be protected.
Question: During Anastasia's ballroom daydream, why is it everyone, except her father, are wearing different outfits than the ones we saw them wear at the beginning of the movie?
Answer: As noted, this is her daydream, not reality. She has a particularly strong mental image of what her father looked like, which is what she sees in the daydream. Also, her father is dressed in a royal uniform reflecting his rank as the czar, so it would be something he consistently wore during formal occasions. As a plot device, it also identifies for the audience that this is her father.
Question: How is it possible Comrade Phlemenkoff never suspected Anya might have been Anastasia the whole time she was at the orphanage? She obviously would've known that the Tsar and his family were killed with the exception of Anastasia, and young Anastasia just happened to be brought to the orphanage around the same time she disappeared and that she's wearing a necklace that looks like its made of gold and jewels. These would have all been major clues that she was the missing princess.
Answer: This was during the Russian Revolution. Millions of people were killed in the resulting violent turmoil, resulting in tens of thousands of children who lost or were separated from their parents. There was no reason to suspect that "Anya" was anyone important and this was a rural orphanage where mainstream news could be spotty. The necklace was likely considered costume jewelry.
Question: Why doesn't Rasputin kill Anastasia's grandmother since she's part of the Romanov family?
Answer: Because she is an old lady, the last of the Romanov line. She was no threat to him, when she died, it would be over.
Answer: As I recall it, Anastasia and her grandmother escaped through a secret passageway when revolutionaries invaded the palace. After the two got separated, the grandmother lived in exile in Paris, where she and Anastasia are eventually reunited. If the grandmother had not escaped, she likely would have been executed. Rasputin probably realised her being a prominent Romanov, as well as a grieving mother/grandmother, could garner public sympathy and outrage over the royal family's brutal deaths.
Question: Why did Anastasia's grandmother wait an entire decade to offer a reward to anyone who could find her granddaughter? This doesn't seem logical since it's clear Empress Marie loves Anastasia very much and is devastated when she gets separated from her by the train. You would think at some point when she got off that train she could've made some effort to get her granddaughter back and not wait so much time to do it.
Answer: She may have tried to find her, but thought she was lost forever. Later in the movie she was getting old and was afraid that her family's legacy would die with her. It was then that she offered a large reward to find her.
Answer: Amnesia exists, but it is a temporary condition. It does not last for the long-term and people usually regain their memory in a day or two, sometimes up to a week. Extreme cases can last longer, but not in the way it is depicted in movies. Some people may lose memories due to severe brain damage from a traumatic injury, but that is permanent.
raywest ★