That’s a lot of questions but I think I understand what it is you mainly want to know.
I changed the 2nd #1 above to be #3 to this won’t be as confusing.
I’m sorry I assumed the neutrals had not been attacked. Something you said in your initial post(s) did indicate that they had been (you talked about claiming Mongolians while Japan was not at war with Russia). I thought you were maybe just confused given the nature of your questions, but now I know I made an assumption I shouldn’t have. In my defense, attacks on strict neutrals by Euro Axis so early do surprise me!
OK, in your situation #1 and #2 have not happened, so you are in #3b territory. The Axis broke strict neutrality elsewhere, so Mongolia has become Pro-Allied, same as Persia or Yugoslavia are at game start.
Russia is not at war with Japan so Russia is neutral on the Pacific map. Mongolia is neutral (pro-allied neutral). As @Krieghund helped us with, Russia is neutral on the Pacific map and neutrals can’t enter other neutrals, so Russia can’t claim Mongolians. But she can merely declare war on Japan (haven’t played much OOB in years, but I think there are no downsides to Russia or Japan declaring on each other)
Japan can attack Mongolia because it is a pro-Allied neutral in your situation, and Japan is at war. Russia may declare war on Japan and then could claim Mongolians like any Allied power at war can claim a pro-Allied neutral.
As far as Japan goes, the rulebook says if Japan attacks any Soviet-controlled territory that is adjacent to any Mongolian territory, all Mongolian territories that are still neutral OR PRO-ALLIES, or have joined the Allies as a result of a FAILED Japanese attack, will be placed under the control of the Soviet Union at the end of Japan’s Conduct Combat phase. Etc.
So if Japan attacks a Mongolia pro-Allied territory in the same Japan turn as Japan attacks a Soviet-controlled territory bordering Mongolia, the rest of Mongolia would instantly become Russian and Russia could move those infantry on the next turn, without claiming them.
Indeed you have an interesting and unusual situation. @Krieghund will audit my answer above, but I think I got it.
I probably should stay out of it since Krieghund is still so quickly responsive, so that Krieghund doesn’t have to sort through my answer and make sure it’s correct, along with your questions.