If by Onna-bugeisha you mean women who were dedicated warriors, then the answer is extremely rare. Tomoe Gozen is the only individul Japanese female warrior to receive any significant description in a Japanese war account (she was described in the Heike Monogatari, an account of the Gempei war of 1180-1185). This account also gives insight into your question about how they were looked upon during that time: Kiso Yoshinaka, the samurai that she served, had resolved himself to death in a dire situation and instructed her to leave, not because he feared for her life so much as he didn't want the dishonor of dying alongside a woman.

Aside from dedicated warrior women, a samurai's family was obliged to serve their lords, in whatever capacity that might entail. While it rarely came to actual fighting, women would often take on the responsibilities of men as the times demanded. They would often engage in repairs, the casting of bullets, and administering medical attention. Another, more grisly duty they often had was preparing enemy heads that had been taken as prizes for viewing.

But, back to female warriors; they were rare exceptions to the norm. There are a number of remarkable stories about women's exploits in the world of the warrior, such as the account of Miyagino and Shinobu, sisters who worked together to avenge their father's death at the hands of a samurai. The warrior women of Aizu is another remarkable exception; during the Boshin War (1868-1869) a number of women participated in the fighting against the encroaching forces of the emperor. The women of Aizu domain were trained in the use of the naginata, and while they were never expected to have to use them, they could. The accounts of the last stand that happened in Aizu include mention of women who fought to the death along side the male soldiers.

I get all of this information from The Samurai Swordsman: Master of War by Stephen Turnbull. It has an entire chapter dedicated to women in the world of the samurai, you might find it interesting. Otherwise, I'll be glad to fill in any more details you might be interested in (as long as I know the answer, of course).

This is going to be a long answer so I hope you don't mind if I address your questions separately.

General History

Onna bugeisha were definitely not common enough to even come close to matching the presence that men had in armies and martial roles. That being said, throughout Japan's martial history, they have been present in various levels of society and at various levels of influence and numbers.

Japanese feudal society was fairly progressive in terms of how women were viewed and what legal rights they had as individuals in society, relatively speaking. Often, women could inherit property for their family, could own and run businesses de jure, and had a voice in social affairs without having to go through a man, whether it be her brother, father or husband.

During the Kamakura and Heian Period, it could be argued that this was the 'friendliest' time for female warriors to make headway in their respective armies.

Much of fighting was still very formal and ritualized.

Battles took place between small numbers of professionals and fighting itself adhered to a large number of ritualistic rules and rigid traditions. Warriors from one side would name a specific person opponent and demand they come forward to face them.

This was far removed from the brutal crush and desperate clashes that characterized the total war of the Sengoku Era, when armies were a mix of conscripts, professional soldiers, and samurai, with thousands of bodies pressing against one another in chaos on the battlefield.

During these times, it is not hard to imagine that female warriors could distinguish themselves on a battlefield, especially when not all exchanges were meant to wipe out the other side. The frantic, brutal grappling and whole sale killing of the Sengoku Era had not become the preferred method of warfare yet.

During these earlier years, the bow was still the weapon by which respectable Japanese warriors fought, competed, and showed their martial prowess. And women could be just as proficient with a bow as many men.

During the Sengoku Period, fighting would have changed drastically and women's roles would have changed as well.

It is no longer a given that female fighters would have a fair chance in a almost ritualistic duel on the battlefield. Neither would any individual man have such a privilege anymore, as the military doctrine of the Sengoku Era shifted to one of win-at-all-costs-total-war. Gone was the reverence for the sacred warrior's fight, replaced by the colder practicality of tactics and strategy.

Why let a smaller force fighter yours one at a time when you can simply crush them all at once?

That being said, women warriors were still heard of during this time. The naginata and swords had fallen out of favor as a competent weapon on the ruthless battlefield of spears, bows and guns. Women would have adopted these weapons alongside their male counterparts.

In general, women of status whose families had a background or tradition of martial prowess would generally receive some martial training but it was far from a standardized practice. It was even more uncommon for a woman to set off with the regular army for war.

If they had any sort of regular role in wartime, it would be as the homeguard, keeping peace at the homefront, perhaps warding off bandits or leading defense in case of enemy raiders. Women of status or influence who had the charisma and determination, as well as popular backing, stepping up to become war leaders in defense of their homes were not that strange. If an able woman took command and people were willing to follow, it would not have been too difficult for men of the time to accept, at least as a temporary measure.

All this being said, women were generally still considered not equal to men, even as warriors. Some famous women are said to have been even greater than their male counterparts as warriors but these are the exceptions rather than the rule. Their status and influence would have been reduced simply because they were woman.

And when Confucian values became popular again after the major wars had passed, women were relegated to a lower status in society than they enjoyed during the relatively free periods of chaotic warfare. Interestingly, it was during this period that samurai women were romanticized. They adopted the naginata as their symbol, much like how male samurai adopted the sword as theirs, despite both of these weapons being considered incompetent for serious warfare (It's important to remember that by the time of the Edo Period, samurai armies had been employing firearms in their regular arsenal for several decades already).

Equipment and Training

During the Heian and Kamakura Period, warrior women would generally recieve a separate training from regular male warriors that took into account their different builds and strengths. That being said, it would share many similarities. The main training for the warrior caste focused around the bow and riding a horse, which eventually led to shooting from horseback, the signature style of samurai fighting excellence.

The women (and men) would have received training in various martial areas, such as fighting with various weapons like the famous naginata, swords, spears, etc. They would also receive training in hand to hand combat. But again, the focus would be on marksmanship, horse riding, and shooting from horseback, whether they were male or female.

During the Sengoku Period some 400 odd years later, the difference in training would be massive. Regular male soldiers, who would fight in much larger armies than their historical cousins, would still have training that was steeped in tradition but focused greatly on discipline and cohesive fighting, particularly with the spear.

Without this, the samurai would be helpless on the contemporary battlefields of their day, where even the greatest warrior could be easily brought down if he faced 4 muddy peasants alone.

Meanwhile, women warriors had very little place on the major battlefields, except for occasionally mounted archers. Their main role would be home defense and maybe fighting off raiders or bandits. Nothing to compare to the great battles the men were fighting.

Their training would be much more similar to the historical training of ritualistic individual combat. It would vary immensely depending on which clan, which family and what region of Japan we were talking about, but in general, training would be more traditional and aesthetic rather than practical. They didn't expect the women who learned some tricks in the dojo to be marching out on campaigns. The main change would be the change in weapons.

The naginata was no longer considered the great weapon it once was, though it still had its place with certain women's fighting styles. Swords were also considered side arms by the regular soldiers but it also had its place.

The key thing to remember here is that for the most part, training women in martial arts was a very decentralized, informal thing and often done on a very individual basis. To contrast, during war time, the vast majority of able bodied men would have had at least some sort of preliminary military training if they weren't already a part of the feudal lord's forces.

Another thing to remember is that during the Sengoku Era, firearms were huge. They were used often and everywhere. Some clans kept them as prestige weapons for only their elites. Others distributed them wholesale and armed entire formations with them. Similarly, sometimes women might have been armed with them, and other times they might have been refused to be allowed to even touch them.

As for armor, women generally would not have a 'female version' armor as they were not represented enough in military forces at any time for it to become standard. Female archers probably would have had a breastplate to protect their breasts from the bowstring when releasing but I do not think we have any actual physical example artifacts from those warring time periods. The oldest one that I know of is from the Edo period. It would have probably looked something like this.

In general, female warriors probably didn't wear much in the way of armor at all, since the vast majority of women who did fight would have been serving as a home defense role and more likely in their everyday clothes more than anything else. The women who earned a fearsome reputation would have had custom armor made for them and like any one off items, they would have been exceptional.