Anastasia (Animation)/WMG

The revolution caused Sophie to lose a few pieces of her mind.
She immersed herself in the culture of Paris for ten years, gaining a French accent in the process, in large part to drown out bad memories. There were also a few other pieces of her mind missing before the revolution. Dowager Empress Marie knew this and kept an eye on her. This, however, is not crucial information. Think about it: Soph had to have strength to have escaped the palace and country, so her peculiarities are (at least as far as the movie plot goes) irrelevant. Evidence and mentions of all of this can be found in the original script which is floating around the Internet. It's basically the same as the movie script we know, but extra bits and characters (such as Sophie's sisters, her opposites) were cut for time.

Sophie is Charlotte's mother.
The timeline is fairly close, they're similar in looks, and Sophie's (presumably) French. Now there's just ignoring the different studios bit.
 * Sophie was part Danish and part Russian in heritage, but she developed a French accent after moving to Paris. Also, Sophie was still in Russia when Charlotte was born.
 * Sophie is not Charlotte's mother then. But, she could still be an aunt, or cousin.

It all is a daydream the real Anastasia had.
The real Anastasia was still a child when she was shot. It was a dream she made up to keep herself happy, about marring a handsome con artist, having her family be safe, wear lots of nice clothes, live happily ever after like in a fairy tale. She was probably too young to realize why she and her family were being persecuted, so she made Rasputin, a friend of the family, the magical villan.
 * The real Anastasia was 17 when she died, but it could still work as an escape fantasy.
 * This has enough Reality Subtext to be very true...and very sad. :(

Sophie is the second chance of Marina Del Rey. Vlad is Benjamin the Manatee.
Watch both Anastasia and Ariel's Beginning with this in mind, and tell me you don't see the resemblances. Marina even had hair of Sophie's color before it fell out. As a bit of trivia, in real life, Sophie's character's inspiration was governess to the Romanov daughters before the revolution, just as Marina was governess to King Triton's daughters. My theory with the animated worlds is that someone believed Marina deserved a second chance at happiness and living a spirited life as opposed to rotting in jail or whatever her final fate was (even if she did become Ursula, which is something people have split opinions over). The mermaid movies end right around the time Vlad and Sophie were born, too. Perfect time for reincarnation.

The movie takes place in an alternate universe/scenario for the Russian Revolution.
This one should speak for itself; in an alternate universe, Lenin and the Bolsheviks managed to not only take the palace down in one revolution, but also managed to have the Czar and (most of) his family killed. In this universe, there is no "October Revolution"; only the February Revolution. Furthermore, in this universe, Rasputin is responsible for the Bolsheviks' takeover, and has supported them from the beginning; he provided them with the funding, spread the word of their cause (maybe he was the one who put up the slogans everywhere), and took over when Lenin was in hiding (possibly receiving Bartok as a present in return). The Czar somehow found out about this, and this was why he called Rasputin a "traitor". Rasputin took this as an insult, as he genuinely thought that turning Russia into a communism would help the country and therefore the Czar should be in gratitude. Angered by this, he finally decided to give the Bolsheviks the signal to attack, and used his little green pixies to help them get in through the gate. And finally, Anastasia was born several years later, which is why she is a child rather than a teenager when all this happens.

Of course, these events all contradict the actual timeline, but when you place these events in an alternate timeline, it all makes sense--the movie is going with that timeline instead of the one in our universe.
 * Fitting into that, Rasputin's curse is what caused the timeline to diverge. He was nearly killed the same way he was in real life but survived and found out members of the royal family were involved (thus his revenge motive, that is true). So he sold his soul to the devil to survive and have the power to take his revenge. His curse caused the February Revolution to work with the help of his minions, but in the process this allowed Anastasia's escape unlike in the real world (somehow, if all else fails, For Want of a Nail can be the cause). So ironically, Rasputin's curse saved Anastasia's life because had he not cast it, she'd have died like in Real Life, not that he would've known that. As for age, like you said, she was just born at a different time.
 * I like this idea. Although the mob in the beginning could be anyone -- there are no red flags or "all power to the Soviets" signs or any similar identification. The dates don't match (Lenin only arrived in Russia after the February Revolution, although it's possible that in this timeline he returned to Russia earlier as well. My proposal is that it's an Alternate Universe where the rioting citizens actually reached the Winter Palace and killed Nicholas and his family on the spot (instead of them getting arrested and executed by the Bolsheviks later). As such, Nicholas had no chance to peacefully transfer power to the Provisionary Government, and Anastasia escaped.
 * That's a pretty brilliant dissection of how and why the two timelines could have diverged. I'm strongly tempted to call this canon.

Pooka is some type of angel puppy.
Anastasia asks for a sign, she gets the dog. Pooka is the one that points out the explosives on the train. Pooka is the one that woke up during the dream sequence to point out she was missing.
 * Alternatively, Pooka is a spy, making sure she stays safe.
 * It's worth noting that a pooka is a type of (mostly) benevolent spirit.
 * Also note he was the one who led Anastasia into the hedge maze--as if knowing she had to face Rasputin once and for all before she could have a happy life?--and that from the start he never liked Con Man Dmitri...only warming up to and accepting him after Dmitri realized the truth about Anya and was going to walk out of her life because he loved her. A lot of insight for just a little dog...

Rasputin's revenge was the result of his Real Life Rasputinian Death
We never find out why Rasputin wanted revenge (that I can recall). The real Rasputin was killed by relatives of the royal family; in this version, he realizes this and makes his Deal with the Devil before he can die, making him pretty much immortal and giving him the means to take his revenge. This explains why he wanted revenge.

Bartok is a folk hero in this universe, and his movie is the story in question.
That explains the anachronisms; it's a 18th-19th century retelling of an old fairytale, or a child's interpretation thereof, and so blends in elements from every time period up to then. Which also explains why he'd end up as a familiar to someone so Obviously Evil. In the first movie, Bartok is either summoned from a "fairytale world" or a homunculus created in his image and personality. After all, who better to serve as an ally then your universe's equivalent of Puss in Boots?