This is a huge topic and probably one I should have addressed this more in text I wrote on historical accuracy because it is one of the biggest changes I made for Choice of the Viking.
I changed the gender identities of a lot of characters from the original texts just so that the game would have far more of a gender balance. Myself, I’m not much of an expert on gender roles in the Viking Age, but it is a large and still-growing area of research, and if anyone is interested in it I would really recommend they read anything by Jóhanna Katrín Friðriksdóttir.
Early Scandinavians and Icelanders have a good reputation in terms of women’s rights. If you google the topic, you’ll probably get a lot of listicles about how the Vikings were actually all feminists because they allowed divorce and women probably did have more rights than in other contemporary European societies, especially those in wealthy families with big farms to be managed (or if they were widows). Still, though, we are talking about heavily patriarchal societies. For example, my impression from medieval stories (and some older poetry for that matter) is that romantic relationships were really all about asserting and maintaining masculine social dominance, with a woman’s marriage as much of a resource as anything else. (The love triangle in Kormáks saga and its very sexy poetry comparing a woman’s cheeks to moons is a good antidote to this, though).
Almost all the evidence says that men held the positions of social power. A goði (pronounced ‘gothi’) was the name for a political leader in the early medieval period in Iceland, but probably also held religious responsibilities as well. It does have a counterpart for women (gyðja), but this word is found incredibly rarely, and, I think, in the texts that remain, a woman who inherits the position of goði has to pass it on to a man instead. I think it has been suggested that gyðja must refer to a more private type of cult (I’m sure someone has used the term ‘domestic’ somewhere in relation to it), but that is harder to back up. The reality is that we have very little information about Old Norse religion, because it is all handed down to us by Christians who kept some bits of information, discarded other bits, and made up the rest. That makes it very hard to be confident about almost anything. Maybe gyðja suggests women had more power earlier in the Viking Age, and there are references to women leading rituals and being warriors.
There is one woman who stands out in the early history of Iceland called Aud the Deep-Minded (aka Unn), but she is remarkable. She was one of Iceland’s original settlers and built up a big power base in the west of the island (not far from where the game is set). It’s hard to say how much her story links with a historical reality, but she is a very important figure in medieval literary tradition.
Topics like homosexuality, cross-dressing, and women taking up masculine behaviours like violence do come up in the sagas and in some poetry, but usually it’s not depicted very positively. This may be misleading, based on the types of sources that have survived, but I can’t imagine that there was a widespread trend that looked on them positively. Homosexuality, in particular, is a theme that has been widely debated and is strongly linked with shame and social dominance. Homosexuality in women is really under-studied in comparison to men-men relationships, I think.
On the topic of Leikny, I’m sorry that I can’t think of much support in early texts for one of my favourite characters. I struggled to find non-binary names, and, as I mentioned, even women wearing men’s clothing is frowned upon in the sagas (and grounds for divorce). Thinking about the myths, Odin and Loki are the characters with identities closest to anything like being non-binary. Loki transforms into animals of different biological sexes without hesitation. It’s probably no coincidence, though, that these are the two classically ‘untrustworthy’ gods.
However, this is also something being studied a lot right now, and I might be behind the curve. There is an interesting archaeological find from a few years ago I could point you to, of a warrior with a possible non-binary gender identity. I haven’t read any of the scholarship on this one, though.
In general, I’m writing all this off the top of my head, and I do know that some researchers are more optimistic than I am on these topics, at least in casual conversation, so I’m very happy to be corrected. As a side note, the podcast Vikings Are Gay goes over a lot of this material as well, is very digestible, and is one of those more optimistic sources.
tl;dr The gender identities and sexual orientations in my game are not historically accurate, but actually, we don’t know much about the past. Check out a book or a podcast?