First and foremost,thank you for asking this question as I never imagined i'd have the opportunity to answer this specific question lol. Luckily for us,believe it or not,a few of your questions have actually been answered here , which I heavily recommend. This picture is pretty famous but oddly enough,not a lot is known about it,specifically its author and painter's identity and its print date. In short, shunga was uniquely both forbidden by law (with further modifications to the law to punish flouts,culminating in the Keikan (sodomy) code of 1872) and widely circulated during wartime,sometimes by military authorities or through seedy pornographic dealers though shunga of the homosexual variety is decidedly rarer.

As for whether the Japanese military were 'aware' of committing sexually violent crimes and to compare and contrast their actions in the Russo-Japanese war with their activities in WW2,you might find em here and in its attendant sources it links to as this question's pretty popular.

As most of my work is already (luckily) done for me,I'll just answer over the remaining gaps that these answers don't really cover ;namely the nature of male homosexuality in Meiji Japan and its treatment (female homosexuality is a whole different ball game and might require a separate question) and whether a nameless Yamato could just buy one of these pieces off the street.

In Furukawa Makoto and Angus Lockyer's frankly amazing piece on male homosexuality (I believe it was the first paper to ever comment on male homosexuality in the Meiji Era),it outlines several "codes" of homosexuality (dōseiai) that defined it during the Edo-Meiji period and I'll talk about the two most relevant to your question.

Note : I apologize for using so many Japanese terms but it's difficult to explain a niche question like this without using appropriately niche terms haha

TLDR : Meiji Japan didn't like the gays,unless they were in the military and it somehow helped them win.

Why are Kyushu men gay ?

The first ideal of homosexuality was Nanshoku (Meaning male lust/sex/colors),which was Edo period dōseiai which involved love between an nenja (an older samurai) and a chigo (younger samurai). It was primarily an expression of samurai ideals,a display of duty and loyalty to older mentor samurais and it was how the idea of male homosexuality was expressed in Japan. Nanshoku was unique in that it did not die out after the Meiji Restoration but evolved to fit the times. Instead of an elder samurai and his protege,it would now take up the form of Senpai and kouhai (Senior and underclassmen). Nanshoku style homosexuality was primarily reported to be in schools,especially the middle and high schools,in which the attendees were male.

Natsume Soseki,the famous writer, once recounted an event that happened to his own brother in a Tokyo school in which

there were still students who had come from the provinces (and) there lingered in the provinces an atmosphere unimaginable to today's youth. My brother told me he had received a love letter from a certain senior,much older than him. How was my brother,who had been brought up in Tokyo and unused to such customs to respond ?

From this,we derive two things. Nanshoku was regarded as something provincial and hicky, something practiced by the people of (as the rumor went) former Samurai domains,like Satsuma. 2nd,people in the big city did not find this normal and at all acceptable. Homosexuality was regional,perhaps a hallmark of "Satsuma culture". The hekonise (samurai/noble children grouped together as part of a boy's club up till their 20th year) were seen as the forefront of this culture and the hekonise system survived in Kyushu and especially Kagoshima (Satsuma Domain) up to the 1900s with the choya shimbun,a local newspaper, commenting on how homosexuality was penalized in Japan reporting in 1886 that "the emotion from loving men was not as any different from that of womanly love"

But,wait a moment here,any self respecting student of Meiji Japan'll know that Satsuma men,specifically its samurai,became the backbone of the Imperial Japanese Navy. So was the Navy allowing (in more ways than one) seamen into its ranks ? (Sorry couldn't resist)

Any idea that has resisted the test of time has evolved to fit with the times and nanshoku was no different. In the military,nanshoku was bushido. Nanshoku was associated with traits like manliness, camaraderie,loyalty and bravery. Which man would not willingly sacrifice himself for his beloved friend ? Which man would not cede all for his beloved brother and comrade ? It became tightly intertwined with Bushido and as Japan scored victories against Qing China and Russia,the rise of Bushido was closely linked with nanshoku culture. At this point,however,the line gets blurred. The IJN wasn't obviously all staffed with Satsuman men but maintained Nanshoku culture. So what was its explanation ? Was the Navy homosexual ? or was it just the Satasuman men who were 'corrupting' the others ? Bishounen no Kenkyuu (Study of Pretty men) by Ozaki Shiro in 1921 solves the Japanese attitude towards this succinctly

"The Japanese Navy dosen't employ those that aren't good looking. The cadets who are employed are loyal and brave. And,so, I recommend the bishonen (pretty boy) of the Navy to represent Japan" In short,it didn't matter whether you were homosexual or not,if being homosexual helped the military continue winning.

I've gone over two (fairly) positive depictions of how Meiji Japan saw its homosexual community but there's actually an important 3rd depiction of them,known as the Kagema model. Without going into extreme detail,kagema were depictions of the stereotypical feminine male prostitute and this applied to male kabuki actors who played female roles as well (onnagata) and were widely viewed with disdain. They were the outcasts,the sissies,the antithesis of everything Bushido professed a man to be. As the kagema population sprung up with red light districts,so too did popular disdain for them. They were regularly arrested when police went on raids,they were mentioned by writers like Ando Kosei to be "weird" and "making him sick" and they were considered a general blight on society. I'd like to take this moment to just say that whether the kagema behaved as luridly as we imagine isn't as important as whether Meiji society perceived them to be degenerate. In the eyes of Meiji society,kagema homosexuals formed the bulk of homosexuals that weren't serve in the military or weren't in school any more and this,unfortunately,shows in their legislature.