1. Anya has NO MEMORIES of her childhood?
Certainly the murder of your entire family is a good reason to repress memories, but at eight years old, you'd probably still be able to remember your full name and the fact THAT YOU WERE A PRINCESS RAISED IN A PALACE ... even if the memories are foggy and no one believes you.
Sure, she hits her head when she falls off the train, but head trauma that includes amnesia would also result in more serious mental injuries. Her fall doesn't look that bad.
Okay, okay, it's just a movie, but it's A LAZILY CONVENIENT SOAP OPERA PLOT TO BASE A WHOLE MOVIE ON.
2. How is Rasputin Defeated by Ice?
Great Ghouly Dark Arts Man overthrew an entire empire but is physically defeated because he made a weak attempt to grab the ankle of a little girl? So very anticlimactic.
Why didn't Bartok just hand Rasputin his reliquary of eternal powers? Even if he doesn't know its true powers, he knows it's important enough to hang on to for a decade after Rasputin's death. BAD SIDEKICK, BAD.
3. Why Does the Dress Change?
The dress Dimitri gives Anya is a lacey billowing green-blue thing with a big ol' collar. The dress she wears while boat dancing is a different shade of blue, no collar, no lace and suddenly channels the Little Mermaid HARD. Dimitri even comments that it looked lovely on the hanger but looks better on, implying the dress was unaltered after gifting (and also that he's totally DTF). It's not some weird consistency issue... it's some inexplicable animation choice.
4. HOW LONG IS HER HAIR?!
In a classic animation makeover fashion, Anya's hair magically triples in length.
The whole "her hair was in a braid-bun" argument can only exist if you don't understand how braid-buns work. (Insider's Guide to Braid-Buns: physics still apply, matter cannot be created.)
5. What Moral High Ground Was Anya On?
With very little convincing, Anya agrees to illegally sneak into Paris, train to be a princess she doesn't really think she is, and then trick Sophie into gaining an audience with the Duchess. After all these deceitful acts, she suddenly gets all high and mighty when she finds out her friends would make money off the con she ALREADY AGREED TO. SHE'S ALREADY DOING IMMORAL THINGS. SHE ALREADY KNEW IT WAS DISHONEST. SHE WAS JUST USING THEM TO GET TO PARIS ANYWAY. Get over the money, girl.