The real Grigori Rasputin was not officially affiliated with the Russian Orthodox Church. He was a religious wanderer, a mystic, and a faith healer, but he was not a priest or a monk. For another bit of mythbusting, Rasputin's influence in politics has been much exagerrated. People at the time thought he was very influential, but that was mostly because they didn't know what he was actually doing. Most Russians were not aware that Alexei was a hemophiliac, so they had no way of knowing that Rasputin was there to heal him.

How much of a belief Russians had in the supernatural is hard to say. But in general, when Rasputin started moving around in St. Petersburg society, there was a fad among the upper classes in mysticism, spirtualism, and the occult. Rasputin was actually introduced to Empress Alexandra in the first place by the princesses Milica and Anastasia of Montenegro. This fad also existed in the rest of Europe and North America. A lot of people certainly seemed to believe that Rasputin had some exotic powers, or that he was hypnotist. But just as many thought he was a fraud. Even the tsar wasn't as reliant on him as most people said. Nicholas kept Rasputin around because Alexandra believed in him wholeheartedly and he believed (rightly) that losing Alexei would probably kill her, too, so he was willing to put up with Rasputin if it kept her happy.

The so-called curse that Rasputin laid on the Romanov prophecy was really more a prophecy in the form of a letter that said that he felt that was going to be assassinated soon and if we was asasssinated by nobles, then Nicholas and his family would be dead within two years, most of the nobles would be exiled from Russia, and that Russia would collapse into chaos and brothers would kill brothers. Which, of course, is exactly what happened. But, really, his prophecy isn't really that miraculous. Pretty much everyone was aware that Russia in 1916 was a powder keg ready to explode, so predicting it took less supernatural powers and more general observational skills.