I don't think the girls were jealous of Alexei. At the time it was natural that he should be the Heir and not them. We are too much used to translate nowadays points of view to imperial girls being born at the end of XIX century and beginning of XX's one.
About little fights and arguments among Alexei and the Grand Duchess, yes...Who could doubt about they argued, and I maybe, quite often? I'm not the daughter of an Emperor, and I fighted with my brother quite wildly time to time!

Nothing odd about it. I think OTMAA were good children, but no matter how good a children is, they argues with his/her siblings and friends. I've already told about Saint Thérèse of Lisieux. When a girl, she was pretty wild, stubborn and not too well behaved. And think she was a Saint! But people thinks that saints are perfect, and in fact, only God actually is...
No problem for me to accept OTMAA as normal children who argues with each other, but loving deeply each other too.
RealAnastasia.