While the Flagellants (Khlisty) had a long prior history, the sect seems to have benefited from a revival of interest and popularity after the Skoptzy (Eunuchs) went into decline in the last quarter of the 19th century. Flagellation as practiced by its adherents (then generally styled Novokhlysty or New Flagellants) was sado-masochistic, as it had been in its earlier manifestations, and their "services" frequently culminated in orgiastic rituals, according to all contemporaneous reports. As Friul noted, it is a virtual certainty that Resputin associated with this group and, as Padre Ambrogio noted in a discussion that he and I lead at CAF, it fostered Rasputin's "strange view of morality, namely that man can be purified from sin only by totally indulging in it, and then living a catharsis of penance and purification". It certainly would have afforded him an introduction to the bizarre mix of asceticism, deviancy, healing, and mysticism associated with his memory.
I think it's also likely that he subsequently had contact with the Novyi Vek (People of the New Age), a sect that would have offered a further opportunity to develop the premises that Padre Ambrogio referenced, since the Novyi Vek refined the cyclical concepts of sin and purification well-beyond the rather crude formulations of the Novokhlysty. The Novyi Vek, also derived from the historic model of the Khlisty, came to being in the last third of the 19th century as the inspiration of Alexei Shchetinin, a Caucasian. He believed that the Holy Spirit would descend on him once he had been baptized by immersion in sin, acknowledged it, and risen above it. This process was repetitive in his view, recurring each time one fell again into sin and subsequently was saved by grace. The ideas espoused by Shchetinen achieved no small measure of notoriety since they invariably involved a plunge into debauchery for a day or more, succeeded by a graced period (termed holy passionlessness), during which the Novyi Vek would concern themselves with prayer, fasting, and good works. When one’s capacity to morally sin recurred, he could anticipate what assuredly must have seemed to be a license to repeat debauchery, safe in the knowledge that it would be followed by re-baptism. Rasputin's alleged visions, prophecies, and healings would have been very consistent with the claims of the Novyi Vek as to the spiritual benefits derived from their excesses.
As to the OP's concern that the picture painted of Rasputin, allegedly a monk, reflects badly on Holy Orthodoxy in Russia of the time, I think it's imperative to understand the climate and historical circumstances that laid a foundation for this kind of thing and to realize that bodies (such as I described above) had little or nothing to do with mainstream Orthodoxy nor with the Old Ritualist/Old Believer movement to which they're sometimes erroneously attributed - since the majority of them arose contemporaneously with the latter.
Although often generically labeled as Bespopovtsy, these sects would have been outright rejected by the Bespopovtsy. They were driven by such considerations as extremism (under the guise of religious fervor), fanaticism, self-aggrandizement, superstition, and zealotry. To my mind, they probably are best classed as Irrational (or Eccentric), Radical, and Extremist sects.
The Irrational (or Eccentric) sects embraced a range of beliefs, frequently mixing secular and religious symbolism, and “different” enough to be classed as “strange”. It’s a generic label for those who shunned the often deadly praxis seen among the Extremists but were, at least somewhat, less theologically bizarre in their tenets than the Radicals. Examples would include: the Dyrniki (Hole-Worshipers) who eschewed the use of icons and prayed toward and through holes in the Eastern walls of their worship houses, watching and waiting for God’s appearance; the Holy Thursday Gapers who kept their mouths open or agape throughout services on that day awaiting spiritual eucharistic sustenance from angels; and, Shepotzi (Whisperers) who perceived the spoken word as a vehicle for sin, but believed that, by whispering (even when praying orally), they could minimize the occasions for sin to which they were exposed.
Those I'd term Radical were principally, but not entirely, millenarian in outlook. Their beliefs and/or praxis exceeded the bounds that would have allowed them to be classed among the Irrational Sects, but, generally, were not so bizarre as to merit being deemed Extremist Sects. These included such as: the Apokalipticheskaya Sekta (Apocalyptic Sect), which set 1666 as the year in which the world would end; the Dvoeverie (Dual-Faith Believers), who incorporated pre-Christian pagan symbolism, praxis, and deities into an otherwise outwardly-Christian worship environment; and, the Napoleoniki (Napoleonites), who perceived the Emperor as the herald of the Second Coming.
Extremists sects were those whose primitive, unusual, or violent praxis and/or beliefs either did not reflect traditional Old Believer ideology or theology or carried it to excesses that surpassed generally accepted norms. Some such groups clearly used the Old Believer movement as a spring-board, from which to promote various bizarre cultic agendas. Most, but not all, of the bodies which centered on extremist behavior, are no longer extant, although rumors invariably abound that secret communities survive in isolation. Many of the most extreme sects were most prominent in the early years of the Old Believer movement; the self-destructive nature of the practices that they advocated virtually assured extinction or devolution into relative obscurity subsequent to ritualistic deaths pursued by their fanatic leaders and principal adherents. Besides such as I already referenced, there were such others as: Bezbrachniki (Free-Lovers) who practiced what has been termed complex marriage in other communal settings, a construct in which every member is theoretically “married” to every member of the opposite sex and exclusive relationships are discouraged, if not forbidden; Duchelstchiki (Stranglers) who practiced ritual strangulation as a means to hasten one’s acceptance into the Elect for those who were elderly or seriously ill, based on interpretation of a text from Matthew that they perceived as requiring a violent death to be assured of entrance into Heaven; and, Ognenniye Kreshcheniya (Fire-Baptizers) who believed that burning christening, as Avvakum had spoken of it, would cleanse them from all earthly transgressions and assure their salvation - the fire in which they enveloped themselves being both their first and last spiritual acts, causing them, at once, to be both enrolled among the elect and entered into the presence of their Savior.
These aberrant ecclesio-sociological phenomena in no way represented Holy Orthodoxy or Old Ritualism, priested or priestless. They originated, in many instances, as a manifestation of the superstitions that were inherent among poverty-stricken, poorly educated persons on the bottom strata of a feudalistic culture. Personification of the Anti-Christ, until then only an intangible scriptural concept, couldn't help but evoke images of all that was foretold in Revelations, including the Apocalypse and Second-Coming. In the more traditional manifestations, there was rejoicing in anticipation of a promised after-life. As one progressed across the continuum, there were other reactions: resigned acceptance of the inevitable end of corporeal life; presumption that these dire events could be forestalled or mitigated through appeasement of the supernatural powers; and, outright despair regarding what was to come. Literalism, eccentricity, fatalism, fear, folklore, superstition - all became contributing factors to the outlook of the adherents and helped to drive their theology and praxis. Somewhere in such a continuum there existed powerful forces ripe to feed zealotry and fanaticism, as people were driven to seek that which would assure (or at least increase the odds of) their salvation or allow them license to do what they would have perceived unacceptable in a society that still had hope of salvation. It was such thinking that spawned many of these sects, particularly the more radical and extreme of them, and individuals such as Rasputin.
Neil, with apologies for (I'm sure) having bludgeoned the transliterations into the English alphabet in many instances