FAQ for
Anastasia (1997) More at IMDbPro »

Even though only Nicholas II and his son Prince Alexey had any claim to the throne, Vladimir Lenin was determined to end the the Romanov line. Along with the that, Russia was in the midst of a civil war, with the "Reds" fighting for Bolshevick side and the "Whites" trying to restore the old order. Lenin, and others, figured that killing the entire royal family would demoralise the White Parisans and eventually cause them to give up because the main reason for their rebellion was now gone forever.

No, he did not. Some documents were found, but were later revealed to be a forgery, written in the 1920s, long after Rasputin and the Romanovs died.

Yes, she did. Back in the 1900s, the dangers of smoking, as well as other drugs weren't well known, and Nicholas and Alexandra even used opium and cocaine when not feeling well.

There was a woman who went under the name Anna Anderson, who from the 1920s to her death in 1984 claimed she was Grand Duchess Anastasia. DNA tests since proved she was not a Romanov, and most likely polish peasent named Franziska Schanzkowska. Her DNA was compared with that of Prince Philip, Duke of Edinburgh whose maternal grandmother was Princess Victoria of Hesse; Empress Alexandra's older sister. It was not a match, nor was it a match to the DNA of the skeletons of the Romanovs, unearthed in 1991. Anastasia did in fact die with her family in Yekatrinburg in 1918.

Anastasia is an animated film from 20th Century Fox, and directed by Don Bluth (Secret Of NIMH, Land Before Time). In short: no.