
Asoda isn't his dad, that would be Akitora and his mum is "Ane-san". The authors other book "Jealousy" shows Tatsuyuki's early life. Asoda is Akitora's right hand in the group (can't recall of the yakuza role has been mentioned) and friend. The theory is that after Ane-san died when her son was 7, Asoda became Tatsuyuki's guardian or at least was the one who was his main carer.

All there is is this bit of dialogue in "Minori's hand" from Asoda, when talking to Tatsuyuki, he replies "Sir, I regret to remind you that our clan's lady (Ane-san/Asami) passed away when you were 7". I don't think Rogi killed her, or at least not deliberately, if he did then he would likely be murdered by Akitora himself. Rogi was also part of the group after her death which seems unlikely if he killed the wakagashira's wife. However, there was a doodle from the author, playing on the cover, which showed Akitora being hugged by both Rogi and Asami, and A is holding a gun saying how she will kill R one day. Rogi as you would expect is just focused on being close to his obsession, not bothered. Basically everything else bar the quote from Asoda is up for speculation :)
I love how parental he is. Tatsuyuki is lucky to have him as his parent. I just adore how he fusses over the boy.