Preved) I'm saying Alexei's existence was "Bizarre" not necessarily "Bad". Serial killers don't have the extreme elements in their life like Alexei had. On one hand a celebrated icon showered with adulation and affection. On the other the victim of often excruciating physical pain and last in a centuries old line of Russian royalty caught in the eye of a revolutionary storm. Serial killers, at best, grow up in anonymous middle class homes, and at worse in hellish conditions. They don't get bowed to by ministers and military leaders one moment, then find themselves laid up in bed suffering from some rare disease the next. When your inner circle includes such as extremes as distinguished heads of state and Grigory Rasputin...and the time line of your life includes such as extremes as being born a Prince in a palace to then dying a prisoner in a dirty basement, in less than fourteen years time, that's how you know you've lived one of the most uniquely peculiar existences in the history of the world.
I agree Alexey's life appeared bizarre from the outside, but probably not to himself. To a child it doesn't matter whether visitors are heads of state or some lowly office workers, they are still "your parents' friends" come visiting and you judge them on how kind and fun they are, not their position in society. His avuncular relationship with all those generals and ministers shows that very well. Not forgetting that many heads of state were family first and foremost, not kings and queens. To Alexey Rasputin was not "Rasputin, the mad monk from Siberia, the horror story that started the Revolution", but "our friend".
Maybe a serial killer grows up thinking his world is the natural state of things, but evidently something is lacking (love), considering what they do later. I am no specialist, but I'm pretty sure many people with different development conditions predisposing them to be serial killers realize that they are different than other people. Often, ironically, as having "megalomania", i.e. they think themselves vastly superior to others (yet feeling intolerably inferior) and thus with permission to harm people they don't like. I.e. conditions which would be very similar to Alexey's actual existence, minus the love that probably cured him of such ideas.