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Im pretty sure you got it spot on.
Abel impregnated 2 women. That would be Reinhard-senseis childhood friend and love, and the japanese woman in the shop. Reinhards lover had 2 kids, which were kain and licht. The japanese girl had tsugumi. Since the japanese girl wanted nothing to do with the family she went back to japan. So the 3 have the same father. This explains why kain also has red eyes like tsugumi.
One thing is off, what do you mean kain met tsugumi? Are you thinking of reinhard sensei? He must have met him when going to japan, or saw him from afar since he was going to kill him, scared that tsugumi would be just like abel and try to kill him. Kain and licht only met him when he entered the school.
this is so confusing.. how the hell kain met tsumugi??
and what the hell happened to the japanese girl that abel impregnated??
i was wondering may be that child of that girl is tsumugi