Christianity and Buddhism are considered pacifist ideologies, do we know why both would produce large and powerful Monastic Military Orders? Have Islam, Sikhism, Hinduism, and other faiths produced similar Warrior-Monks?

So, yeah, this is kind of tough to tackle. Not because the question itself is particularly hard to answer, though there are some major nuances to it (Islam and Sikhism have no monastic orders and the Hindu one was founded about a thousand years after the Buddhist and Jain ones in direct --if belated-- response to them), but because the premise it self relies on a very western and very modern assumption. While it is true that Buddhism has a strong component of Non-violence to it, much like Christianity, this is often at odds with the political and social reality of the adherents. Their positive non-violent conceptions come from modern day activists such as HH 14th Dalai Lama and Pope Francis who have both promoted pacifist interpretations of their religious traditions and been successful on a world scale (kind of, the Dalai Lama may be the most admired person in Europe, India, and America according to polls, but is reviled in most of China and major parts of the Muslim world), but especially in the west. Islam, Sikhism, and Hinduism each have pacifist traditions but since they don't have as much of a presence in the west, we don't often see them as such.

Buddhism especially has a perception of being entirely monks meditating on the self and chanting, praying for peace, but it's actually not the most important tenet in Buddhism (which isn't to say that it isn't important). Jainism and many forms Hinduism take pacifism even farther with Jains going to extraordinary lengths not to harm life of any kind, avoiding even harming micro-organisms.

With that out of the way, to answer your question, and I'm going to only answer the first part from a Buddhist perspective, warrior-monks in Buddhism are quite rare in it's 2,500 year history. And, unlike Christianity, I don't believe there has ever been a centralized order of them but rather groups of Monks and Temples acting individually for self-defense or "defense of the religion". This isn't to say that wars and crusade-like events haven't been religiously motivated in Buddhism, far from it; Tibet, home of one of the largest promoters of Non-violence and Buddhism, was shaped by many such conflicts during the fall of the Tibetan Empire and the re-unification of Tibet under HH 5th Dalai Lama and Gushi Khan. even today, there are Buddhist extremists fighting religiously motivated conflicts (some might say ethnic cleansing) in Myanmar and Sri Lanka.

As to why such orders were formed, it came down to social factors. In China, Ch'an (Zen) monks were taught to be self-sufficient and martial arts were for many temples part of this practice, as well as a way of meditation. At many times during the history of Ch'an, there was social upheaval and government persecution, especially under the T'ang which forced monks to need a way to defend themselves both from raiders and government forces. In times of both peace and war, they continue the practice of martial arts, though warrior-monks may be an extreme title given that they are essentially just monks that know martial arts.

To my knowledge, there has never been a formal Buddhist military order that was disconnected from a national government. I guess we could count Zen monks during the Russo-Japanese War and WWII since they promoted a call to arms, but they never participated directly in the conflict and the motivation was very different from a monastic military order.

The answer to the second half of your question is basically yes. The initial heads of Islam (Muhammad and his successors) were all religious crusaders as well as preachers and heads of state. This shaped the early history of Islam and has had repercussions for its unfortunate perception as a violent religion. I don't feel comfortable expanding upon why or the exact structure of this, since it isn't my area of study and with Islamophobia being so prevalent in the world right now, I don't want to risk saying anything wrong and inflammatory.

As for Sikhism, while it doesn't have warrior monks per-se, many of it's Gurus were also warriors due to discrimination from the Mughals (Jahangir and onward). In 1606 Jahangir captured and tortured Guru Arjan to death after his refusal to convert to Islam leading to further persecution by the Muslim leaders in the Punjab. In 1675 the ninth Guru, Guru Tegh Bahadur, was publically executed under the orders of Emperor Aurangzeb officially for extortion, but the Sikhs thought it because of his refusal to convert and the growing influence of the Sikhs (though it should be noted that he and the previous gurus supported Aurangzeb's brother Dara Shukoh in a war of succession due to his moderate Sufi tenancies). Whatever the reason, this became a rallying cry and Guru Gobind Singh established the Khalsa, a warrior community of Sikhs who were baptized and took oaths, this was not a community of warrior monks, but a tradition of active resistance by the devout. The Khalsas established the Sikh confederacy and later the Sikh Empire and to this day have an active division within the Indian Army.

Hinduism is again more complicated. The Vedic religion was stratified by Varna (though how much this was put into practical use remains a subject of some debate) where the Kshatriya class were warriors and leaders and the Brahmans were priests and officials (though at the very least there was mobility and some Brahmans did become soldiers and kings). As the Vedic religion gave way to the beginnings of modern Hinduism, the social structure appeared to be again more flux and there was no central religious authority. The thing we have to realize about Hinduism is that it is really a plurality of religions and even centralized organizations, like monks, are comparatively recent. This decentralized religious structure made religious crusades difficult, though they did happen, it wasn't under a pope or centralized religious authority, but rather under a king or general. We also have to note that there was a general tolerance for other religions throughout Indian history with most kings and rulers supporting ALL religious traditions as a matter of principal, thus we see massive Buddhist temples built with the money of the Vaishnavite Guptas.

So yeah, it's kind of a difficult question to answer not only because the premise is somewhat flawed, but also because we are coming at this from a Western perspective and all the baggage that comes with that.

Sources:

Robinson, Richard H., and Williard L. Johnson. The Buddhist Religion: A Historical Introduction. Wadsworth Publishing Company, 1997.
Singh, Khushwant, and Pratap Singh. A History of the Sikhs, Volume I: 1469-1839 (Second edition). Oxford University Press, 2015.
Chen, Kenneth K. S. Buddhism in China: a historical survey. Princetion University Press, 1973.
Laird, Thomas, and HH 14th Dalai Lama. The story of Tibet: Conversations with the Dalai Lama. Atlantic, 2007.
Ganeri, Jonardon. The Oxford handbook of Indian Philosophy. Oxford University Press, 2017.
Patrick, Kit. History of India Podcast. 2015-present.
Lewis, Bernard (1995), The Middle East: A Brief History of the Last 2,000 Years. Scribner