I haven't seen the 'Anastasia ' cartoon, (nor do I wish to see Saints of my Church portrayed in cartoons), but I imagine the 'Rasputin' in it fits in pretty much with the image that has been made of him. No doubt he did much that was wrong, maybe much of it unintentionally, but apparantly he had a strong sense of repentance. I do not want to make a saint out of him, but there is so much rumour and innuendo about the man it is hard to know what is true - it was true back before the Revolution, and those rumours continue to fuel the popular image or idea of Rasputin.
From what I have read, the Tsar's children all liked him, (though in fictional novels you will always have one of them hating him, or thinking he is evil or whatever, and why don't Mama and Papa realise, or my sisters, but the reality was that the imperial Family's image of Rasputin was very different from the popular image, much of which was exagerrated and fed by rumour and gossip.)