Hi Jenjen--
These questions have all been asked and responded to previously on this forum, but I can't blame you for not knowing that, especially since most of the categories have pages and pages of threads, and most of the threads are quite extensive. I'll do my best, then, to encapsulate my own thoughts on this topic, and perhaps others will add their own thoughts as well.
Alexandra had a very "typical" relationship with her daughters. What is "typical"? Well, it included many ups and downs, but it was loving and close, and she and her daughters were definitely bonded. Unlike many royal mothers, and indeed many Victorian/Edwardian mothers of almost station, Alexandra was a very "hands on" mother. She breastfed her daughters and she was with them as much as possible during their growing-up years. Read the letters of her daughters and you will note that she playing games with them and that they frequently spend both leisure and public time with their mother. This isn't to say that they were with her 24/7. Nurses, nannies, governesses, tutors and--ultimately--ladies-in-waiting also played a role in taking care of Olga, Tatiana, Marie and Anastasia. But Alexandera was very much a prescense in the day-to-day lives of her daughters. Some, in fact, would say too much so . . . and in some instances, I would say they were correct.
Olga was the eldest daughter, and I know from my own experience that oldest daughters do not always have an easy time of it with their mothers, especially if those mothers are more-than-typically protective. But there is much to indicate that Olga's relationship with her mother, while at times troubled, was not seriously threatened to the point of irreparable chism. Read Olga's poem to her mother, written while they were in captivity, and you will see that Olga understood her mother and loved her very much.
Alexei, being the heir, would of course be the focal point of the family. And, as you point out, his illness also made him a center of concern. I think all four girls accepted and understood this, though of course all siblings have complaints about each other at times, and it had to have crossed their minds on occasion that such-and-such wasn't fair. But such thoughts probably would not have lingered, for after all it was a male-centered culture, and the sensibilities of 100 years ago are not the sensibilities of today.
Much has been made of Marie's sensitivite nature, and I would agree that as a "middle child" she no doubt had issues related to that fact. I don't think, however, she was aware of her birth being "a disappointment." Most of us know about the initial reaction of her father to her birth, but we also know that Nicholas loved Marie very much, declaring her to be the family angel. As for Anastasia, I would not doubt that much of her bumptious personality was related to the fact that she was the fourth girl, followed by the only boy, and therefore had to do SOMETHING to get attention. But again, I doubt she was intellectually aware of her birth being a "disappointment."