I'm going to focus on the latter half of your question, specifically about female warriors in this time frame since a full accounting of Norse views and expectations of women could, and has, filled several books. However that is not to say that the issue of female warriors is any less complex.
The issue of women being warriors in Norse society is fraught and controversial to say the least. There isn't a clear and solid scholarly consensus on this subject at all, and it has only become more clouded with recent archaeological discoveries, in addition to debates surrounding the limited textual evidence. This might seem a little counter intuitive, if a source, for example John Skylitzes, mentions female warriors in a society that is partly composed of Norse descended elites, that should be relatively open and shut, right? Unfortunately that is not the case. Many potential biases, misinformation, and other such problems can prevent us from accepting primary sources at their word.
It has become a quite popular trope in media depicting the Norse/Vikings, think Lagertha's character in History Channel's Vikings, but also fictional societies inspired by the Norse or other Germanic societies, think Skyrim's warrior women such as Aela the Huntress, Lydia, and so on, as well as examples such as Tolkien's Rohirrim with Eowyn.
So where can historians, and archaeologists, turn to try and sort out this debate? Relying on mythic tales such as the Valkyries is unhelpful to say the least. Norse mythology, artistic depictions, and so on are not a substitute for historical accounts or firm archaeological evidence. Further complicating the issue is the fact that many sources on the Norse and their ways of life and beliefs are either from outside perspectives or come well after the Christianizing of the Scandinavians and the subsequent cultural change that this brought about. I don't mean to imply by this that Christianity is what ended the existence of viking female warriors and I've yet to see a serious argument that proposes such.
As I mentioned above, there are some textual accounts of women fighting in Norse/Norse derived cultures, and I'm going to go through a few such examples and provide some added context to them.
The Byzantine chronicler John Skylitzes wrote in his history of the Byzantine (Roman) Empire about the myriad conflicts and enemies that were faced by the Byzantines. During one attack by the Kievan Rus (I believe its Varangians in the original text) into Bulgaria, he mentions that in the aftermath of one battle, the Romans discovered that among the enemy dead were women wearing armor and fighting alongside the men.
At this point in the wars between the Byzantines and the Bulgarian/Rus the Kievan king was in quite desperate straights, indeed shortly after this incident he would be defeated and set back to his homelands after signing a treaty with the Romans. So while it would be speculative to say that women fought only under these desperate circumstances, it cannot be dismissed out of hand. It is also worth bearing in mind that the chronicle himself is intensely biased against these people. He describes their lamentations as sounding closer to beasts than men, among other colorful descriptions. Furthermore these events took place in the late 10th century, 971, and half a continent away from England and Scandinavia.
Leo the Deacon also wrote about this campaign in his own work, and as far as I am aware he made no mention of women fighting alongside their fellow men during this campaign. Skylitzes was also writing nearly 100 years after these events.
Saxo Grammaticus, a Danish chronicler, wrote in his Gesta Danorum about women who also fought alongside men and even had their own armed retainers and went into battle eschewing armor. He also mentions that one woman fought and defeated a Swedish champion before falling to arrow fire.
However the issues of distance are extremely pronounced in this case. Saxo Grammaticus was writing centuries after the events he described about a largely mythic past. He also mentions many fantastical and, as far as I am aware, uncorroborated aspects of Norse warfare such as the use of chariots.
Both of these accounts come quite removed from the events that they are describing. Skylitzes was after all a Greek historian describing a war that had concluded decades before he was born. Saxo Grammaticus was indeed Danish, but he was writing about events that no corroborating textual history goes back to. Issues of the fetishization of textual record aside, there is very little to corroborate what he says, and given the fantastical nature of the events that he describes, more than a little skepticism is necessary.
What about sources such as the sagas? While written far removed from the events depicted, many of them are based around historical events and some, such as the Vinland saga, ultimately have been borne out by archaeological evidence. Indeed in the sage of Erik the Red, and the related Greenland Saga, Lief Erikson's sister is mentioned a fierce warrior. In addition to the relatively corroborated stories relayed in these sagas, there are also numerous warrior women who appear in the sagas, however many of these are much more explicitly mythical and lack the relative grounding of the ones I mentioned above.
However it is clear than in these stories, there does exist a tradition, grounded in reality or not, of women who fight and engage in other more traditionally masculine pursuits. When combined with the outside accounts it is very tempting to bring this all together and say that there were indeed women who partook in warfare in the Norse world.
However, despite how convincing all of this textual evidence may seem to be, there are scholars who disagree. Judith Jesch is a very prominent example of such a scholar and she has become a little more well known for her skepticism, as evidenced on her online blog as well as in other writings, about the idea of Norse warrior women.
Last year there was a find in a Swedish archaeological site in Birka that the inhabitant of what was considered a traditionally masculine coded grave was actually a woman. This set off a storm of online activity, and I'm sure most people here can imagine the sort of clickbaity articles that were spawned immediately.
Here is a link to the original article http://onlinelibrary.wiley.com/doi/10.1002/ajpa.23308/full
and here is one to Professor Jesch's blog post in response to the article http://norseandviking.blogspot.com/2017/09/lets-debate-female-viking-warriors-yet.html
Both of these are well worth the read.
The unfortunate truth is that there is not any sort of scholarly consensus on this topic and it is still hotly debated. I've given some examples from textual evidence and a recent discussion about the archaeological side of things, and I hope that helps you out. This topic is sadly not conducive to a flat out yes or no question
Sources and further reading
John Skylitzes, Synopsis of Histories
Saxo GrammaticusGesta Danorum (Book 8)
An excellent overview, particularly for the first part of your question which I skipped to focus more on the warrior woman aspect with be Jennyy Jochen's Women in Old Norse Society
I mentioned Professor Jesch's blog, but I would be remiss to not also include her own book on the subject of women in Norse society, Women in the Viking Age