all three organizations are considered to be fundamentally Christian. In the case of the Templars, again thanks to the Troyes Fraternity and St. Bernard of Clairvaux, most historians agree that the organization was Christian, yet not Roman Catholic. Cistercianism traces its roots much more to the Essene, an isolationist civilization in the Jewish Levant that seems to have finished at about 40 AD

Where on God's good earth did you get the idea that the Cistercian Order traces its roots to a Jewish apocalyptic cult from late antiquity? They're a reforming monastic order that grew out of the so-called twelfth-century reformation along with a whole host of other more austere takes on traditional Christian monasticism. Basically, groups of Christians decided that the traditional/Cluniac/Benedictine monastic life had grown too opulent for monks (and nuns) to fully extricate themselves from the pleasures of the world to focus on Christ. The founding ideals of the order involved the freely given, knowing consent to become a religious (as opposed to your parents donating you as a kid), the value of manual labor (this...didn't really last, or rather, they delegated the work to semi-monks), and stripping the ornateness out of church and liturgy to distill it down to the bare essence for their life of ceaseless prayer. Medieval Cistercians attended and (ordained monks only) performed Mass and sang the Divine Liturgy, and the order's patron saint from early on was the Virgin Mary--you know, Christ's mom.

I guess you could make an argument that the military monastic/ecclesiastical knighthood orders aren't "Roman Catholic" in the strict sense, because until the Reformation and Trent there was no "Catholic Church" in opposition to western Protestantism. But "catholic Christians" is a term already thrown about by Augustine, and the surviving orders/their descendants (like the Order of Christ in the early modern Kingdom of Kongo) are absolutely part of the Catholic Church in the continuity of the Church answering to the authority of the Bishop of Rome.

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Of the three great military orders (the Templars, Hospitallers, and Teutonic Knights), it's difficult to settle on an overall winner in terms of importance. The Temple loses out by having a lifespan some seven centuries shorter than the others. Also, importance is quite difficult to quantify. It is easier, however, to discuss what sort of importance they had across the time period.

Until the mid-thirteenth century, the Templars and Hospitallers were both noticeable political and military powers. The Temple was the oldest military order, founded in the early twelfth century. The Hospitallers were older, dating from the late eleventh century, but they didn't militarise until later. Both orders played a major role in the defence of the crusader states and both had lands and brethren across Europe. As the Order's European estates annually sent money to the Holy Land to support their brethren there, the Templars became adept at transporting goods and cash abroad. This led to them providing banking services for European nobles and monarchs. The Hospitallers did the same, eventually, but it was only after the arrest of the Templars in 1307 that this became a more important role for the Hospital.

As significant landowners in Europe, the military orders, like most religious orders, began to be relied upon by secular lords and kings. In England and Ireland, the Templars were initially the main military order to act as royal servants, undertaking financial and diplomatic work for the kings of England. For example, the master of the Irish Templars was a regular auditor of royal accounts, whilst in 1234, Templars led negotiations between royal forces and the rebel Richard Marshal, earl of Pembroke. However, by the late-thirteenth century, this position had been taken by the Hospitallers, who by the end of the medieval period were regular royal councillors. The Prior of Ireland, the Order's master in Ireland, served in various high offices, including chancellor and justiciar, the two chief administrators of English Ireland. One Hospitaller, Stephen de Fulbourn, was treasurer of Ireland in 1274 and justiciar from 1281-8. The Prior of England, master of the Hospitallers in England, Wales, and Scotland, became the premier baron in the English parliament, making him the most senior lay lord below an earl. He could also be a close personal councillor. Prior Botyll was among the first councillors allowed to see Henry VI after he recovered from his first bout of madness.

Both orders had a pan-European appeal. The Templars and Hospitallers had lands in Germany, France, Italy, Iberia, the British Isles, and the Holy Land, and brethren from the same. In contrast, the Teutonic Knights were rather minor to start with. This order was only founded in the 1190s, and so at first lagged behind the more senior Hospital and Temple. The Teutonic Order, as its name suggests, was also more limited in scope, recruiting donations and membership mostly in 'German' lands. The Order established itself across Palestine, the Low Countries, and what is now Germany and Austria, and held some land in Sweden. However, regions like Iberia, France, and the British Isles did not have Teutonic outposts and seem to have had little contact with the Order (though the Order did trade with these countries and host crusaders from them). It is only when the Teutonic Knights began to conquer territory in the Baltic in the thirteenth century that they seem to become a real force. This made them one of the important orders, as no other military order had established its own independent state.

After the Hospitallers conquered Rhodes in around 1310 they joined the Teutonic Knights as one of the two most militarily and politically important military orders. They were the only orders to maintain their own independent states, the Hospital in Rhodes, the Teutonic Knights in the Baltic. In the Mediterranean the Hospitallers always remained a key feature in crusade plans. Their conquest of Rhodes from 1309-10 was itself a papally-approved crusade, with the aim that the island would form a bridgehead for a second expedition to retake to the Holy Land. Meanwhile, in the Baltic the Teutonic Knights continued to grow in power. A system of summer raids called reisen became what can be almost described as a medieval tourist attraction. Crusaders would visit from all over Europe to feast with the Knights, participate in tournaments, receive prizes for valour, and fight the pagan Lithuanians. Famous participants include the Douglas earls of Scotland, Henry IV of England, and Jean le Maingre, marshal of France. However, after the Order's defeat by Polish-Lithuanian forces at Tannenburg in 1410, participants for the reisen dropped off sharply, with the last foreign visitors, a group of Burgundian knights, arriving in 1413. The Lithuanians had long since converted to Christianity and with their alliance and monarchical union with Poland, this could no longer be ignored or considered insincere. The Teutonic Knights had lost their prime purpose of holy war. Attempts were made to recover this by relocating to Translyvania to fight the Turks, something the Order declined (the offer was conditional on giving up Prussia), or by going to war with the schismatic Russians. This second path was fatal, and eventually brought about the end of the Ordensstaat.

These are very broad questions. I would suggest narrowing your search. With questions this broad everyone will be an expert, and you will get far too much opinion.

However, I love that the middle ages can fire people up this much still, online.