What does traditional Finnish thinking say of gender? The modern word is sukupuoli, literally "family side" or "sex side", or a "half" specifically. This lines up with dualism in Finnish tradition. How this differs from the usual thinking is that Finns also believed it was fully possible to cross the border between these two sides.
In the early 20th century, an Iron Age grave was discovered in Hattula, Häme. This burial is then over 1,000 years old. At the time, it was seen that there were men's burial items in the grave, such as a sword. However, the only body discovered was wearing women's clothes and jewellery. At the time, it was assumed that there had been two people originally buried there, a man and a woman, but only the woman's remains had been found.
Further research showed there was no one else in the grave. Decades later, the narrative had shifted: the one in the grave must have been a warrior woman! You can still find claims of this online, of a sword of a warrior woman from Finland discovered. However, modern DNA research has proven otherwise.
The individual buried was a "biological man" with Klinefelter syndrome: they had XXY chromosomes. This causes certain "womanly" features to develop, such as breast growth and wider hips. From the grave, it is obvious: while we can't know the gender identity of this individual, they certainly surpassed the border between male and female, as the burial proves. This cannot be a case of humiliation either, as the individual was buried lovingly, with care and precious items.
"We will never know what the ancient deceased thought of their gender, but based on the burial artifacts and sex chromosomes, a simple divide between male and female did not fit them, and it looks like their community had no problem with that." (https://yle.fi/a/3-12030116)
It wasn't that uncommon to assume the crossing of this gender border either. If there was a masculine woman or a feminine man, it was often assumed that this person could (or even "had to") be kaksineuvoinen, or intersex. It seems that people also assumed this condition was more common than it actually is. Quite the downfall in modern Finland, where the laws are not ideal regarding intersex people. Still, I can't possibly emphasize more how, to Finns in tradition, not fitting into male and female gender norms was completely okay. It's not the only border that can be crossed, after all. For example, the border between life and death, this world and the other side, is crossed all the time by shamans! Therefore, with a little bit of magical aptitude, crossing the border of gender is in no way impossible.
Sexual orientations are a whole different matter. People did not traditionally have the concept of sexual orientations, so identities based on those did not exist. As Finnish deities are mostly personifications of natural phenomena, they don't really have sexual orientations because... why would you ask what is the sexual orientation of the Moon, it's just the Moon bruh.
With this, we have to instead turn to look at what folk tradition mentions about same-sex couples and sexual activities. There are no specific terms associated with such identities or actions, which implies that while it wasn't an identity, it wasn't something forbidden either. You'd need words for a thing to forbid it, after all. It didn't seem to have any stereotypes or classifications attached to it, and possibly most people just never thought about it.
There have been little stories and jokes about two old maids, or young girls, who have satisfied each other sexually with dildos or lyömällä lattapillua (scissoring? idk man...). This stuff was mostly told by men and the jokes tried to "prove" that this would've all been better if a "real man" would've been involved. There are also jokes about, for example, two men ending up on a deserted island together. Still, homosexuality was not shown in the kind of negative light that zoophilia, necrophilia and incest clearly were.
It is fair to say that: "there was no word, no theory, nor a category for sexual variation like homosexuality". You might run into the claim that an old word for a gay man in Finnish was miehimys, but this term originally meant an aggressive horse or a sexually eager (towards men) woman, and the meaning of a gay man was only added somewhere around the turn of the 20th century. Only descriptions like "he is always attached to others' balls" existed. As stated earlier, when it comes to feminine men and masculine women, there was no assumption of "gayness".
Agrarian society was focused on building a family and having children to continue the means of living. While unmarried people were very useful and helpful in society, they were often mocked, though I have to emphasize there was no idea that you needed to get married super young (like in the 30s was completely okay). Romance was not valued much, and public displays of affection were seen as a ridiculous thing of the upper class, until romance made its way into folk music in the 19th century. The possibility for homosexuality became more of a thing when the unmarried population of the villages could move into cities, and the position of such a person was way more limited on the countryside, though many said that if there were two men or women living together, they were not thought of in any kind of special way.
There are stories of individuals, married men and women, who decided to get sexual fulfillment from other men or women instead, and this caused discord in their marriages. The reaction is negative precisely on the perspective that they were failing their marriages, not the same-sex relations in and of themselves. There are also descriptions of lesbians on the countryside: "women who were interested in women, had developed a crush on a woman and even proposed". Reactions towards homosexuality were then mostly neutral, though could've been mildly negative causing ridicule, but not aggressive. Markku Koski stated that there was no moral panic over homosexuality because that would've required the defining of the "correct" kind of masculinity and femininity that just simply wasn't there. Similarly, there was no idea that the sexuality of men was "dominating" and the sexuality of women was "passive", so there was no possibility for say, a man's sexuality to be "inverted" as a bottom in same-sex relations. This differs quite a bit compared to what I've read, for example, about the "Vikings" or Roman Empire.
Summa summarum, the modern stereotypes of "gay people" only became a thing in Finland after the Second World War from the upper class and the newly formed middle class. I see, then, nothing in the Finnish tradition or paganism that would encourage negative attitudes towards minority sexual orientations or gender identities. The only thing that many, myself included, would have a problem with on this front is the absolute focus on reproduction, and disapproval of remaining unmarried, but I personally see those as symptoms of the agrarian societal structure that we obviously don't live in anymore.
PS. Runosongs don't really refer to this topic, though that is not surprising as depicting romance became more common only at a later point. Still, I have found this love incantation written down in 1903 which seems to include some queerness:
 Ei oo yksin hyvä olla Armas neitinä eleä Syty siis, sä lemmen tuli Syty väkevä, voimakas! Syttyi myöskin Väinämöisen Rohkean laulajan povehen Nähdessänsä neitosia Katsellessaan kassakoita |
It's not good to be alone Not lovely to live as an unmarried miss So flare up, then, you fire of love Flare up, you strong and powerful! It also flared up in Väinämöinen In the chest of the brave singer When he saw maidens When he looked at horsemen |