My opinion on Rasputin was that he was very, very canny and smart (which is different from being literate and good at memorization) and his "prophecies" were the result of nothing more than his ability to read the writing on the wall where the Imperial family was often too self-involved, naive and misguided to do so themselves.
His warning to the Tsar about the terrible things that would happen if a member of the Tsar's family killed him (presuming he actually made that prophecy) seems more like a threat to me. He knew the family wanted him dead, and he was telling Nicholas to make sure his family backed off. The fact that his "prophecy" came true is not surprising - there are tons of people in 1916 who could have predicted doom for the Imperial family, even if they themselves were too out of touch to realize it.
As for him healing Alexis, I always read that his major impact was 1) He got the doctors to stop poking, prodding, examining and constantly moving Alexis, which allowed his blood to clot and 2) He calmed down both Alexis and his mother, and the massive decrease in tension and hysteria also allowed Alexis to heal. The rest? Luck. The same reason other hemophiliacs survived attacks.
I would like to know more about his relation with Alix, because i had red one year ago that had been found same letters that Alix had sent to Rasputin and that could show that they had had any sentimental relation, and i would like to know if this information is truth or false.
Alix's letters to Rasputin were published back then by enemies of Rasputin and they are very sentimental. In Robert Massie's book Nicholas and Alexandra, he reprints the most "damning" of the letters, but points out that Alix wrote in that style to everyone. Here's a snippet:
I kiss your hands and lean my head on your blessed shoulder. Oh how light, how light do I feel then. i only wish one thing: to fall asleep, to fall asleep, forever on your shoulders and in your arms. What happiness to feel your presence near me.
It does sound like a love letter. It ends with "I love you forever". I don't believe there was anything sexual going on, but I do believe Alix's attachment to Rasputin was abnormal and unhealthy. She even had the Grand Duchesses wear his picture in lockets around their necks. The Russian people were completely right to think that the Imperial family showed very poor judgment in following Rasputin to the degree they did.