Huh. Feels a bit as if the author watched desperados, or other older action movies, and tried combining them with japanese setting.
Also, while there are workarounds, unless im wrong protestants do not do confessions to priests, while catholic priests cant marry/have kids (just a small note, who cares in a manga)
Traditional rite Anglicans and old rite Lutherans still have confessions. That said, there are only around 500 members of the old rite Japan Lutheran Church, so this is probably not what's being portrayed here. The Anglican Episcopal Church in Japan, on the other hand, is IIRC the second largest Protestant congregation in the country (after the United Church and if not counting the ten or so Baptist groups as a single denomination) and they're VERY traditional rite - so much so that you'll find many concordances between them and the much larger Catholic Church in Japan. Japanese Anglicans are practically Catholics in rite and liturgy, but liberal in outlook and theology. While part of GAFCON, they actually aligned with Canterbury on most items on contention, including the ordination of women and tolerance (but not acceptance) of homosexuality. For that matter, Primate Nathaniel Makoto Uematsu's replacement as bishop is a woman. Nevertheless, they accepted the recent GAFCON/Global South resolution to depose Canterbury as the Primus Inter Pares of the Anglican Communion.
So, given the state of the two Protestant denominations that practice confession in Japan, that comment about homosexuality suggests this is the Catholic Church.
For all intent, he could have had the daughter before becoming a priest. There's no objection in the Catholic Church to a widower becoming one.
As for reporting an assassin, I think it depends on the laws of the country they're in, not on a particular Church policy. If they don't report, I think they'd normally encourage the person to turn themselves in.
The formal position of the Catholic Church has always been - in the words of Pope Francis himself - that ordination is eternal but clerical celibacy is a matter of discipline. If you fail at being celibate you get, well... disciplined. That's why the pedos didn't get defrocked but just exiled. If a priest fathers a child after ordination, he's obligated to raise that child. That said, this case is probably more like what you say.
Murder is not absolvable by confession. Church policy is to always report a confession of murder to authorities. The catch is that canon law defines murder very specifically as the unjustified, "wasteful" (meaning purposeless) killing of an innocent. The Church does accept that some forms of killing are justified and absolvable - especially when committed in the name of God. This is after all the legal basis behind the indulgences given to crusaders and the more violent agents of the Church (including professional assassins employed by the Vatican). My take here is that the priest already figured out (because it's pretty damn obvious just from reading the news) that Rose only kills criminals, so he marks her deeds as justified killing instead of murder.
In Italy traditionally this was the same kind of rationalization made when mafiosos confessed to priests. Most crime organization violence is directed towards other criminals, so there was even a time when a priest would warn confessing mafiosos to only confess intra-underworld violence. This attitude changed drastically after the Falcone incident and the Church itself started snitching to the cops.