See The Medieval Origins of the Legal Profession by Brundage:
> In the aftermath of sixth-century barbarian invasions, the legal profession that had grown and flourished during the Roman Empire vanished. Nonetheless, professional lawyers suddenly reappeared in Western Europe 700 years later during the 1230s, when church councils and public authorities began to impose a body of ethical obligations on those who practiced law. This book traces the history of legal practice from its genesis in ancient Rome to its rebirth in the early Middle Ages and eventual resurgence in the courts of the medieval church. By the end of the eleventh century, the book argues, renewed interest in Roman law combined with the rise of canon law of the Western church to trigger a series of consolidations in the profession. New legal procedures emerged, and formal training for proctors and advocates became necessary in order to practice law in the reorganized church courts. The book demonstrates that many features that characterize legal advocacy today were already in place by 1250, as lawyers trained in Roman and canon law became professionals in every sense of the term. It also examines the centuries-long power struggle between local courts and the Christian church, secular rule and religious edict.
* https://chicago.universitypressscholarship.com/view/10.7208/...
* https://utpress.utexas.edu/books/perlman-ancient-greek-law-i...
From the little I know about Islamic jurisprudence, specifically through the works of Toby Huff,[1] the tribunals were more 'private' and the rulings were done more by something that we'd call arbitration, and 'case files' would remain in the judge's/arbitrator's possession.
In roman times, if someone killed your cousin you could go to court and it would mostly end there. After the collapse, if someone killed your cousin your family had the right to kill someone of "equal value" from the family of the killer. Aside from the brutality and senseless violence (as seen from us today), the biggest problem was that people disagreed on "equal value", resulting in a feud, i.e. an escalation of revenge because each party felt that the other side's killing was way more than warranted to make things even.
So the states (starting from the early barbarian kingdoms) would encourage the family of the killed to accept a monetary payment to avoid the murders to spiral out of control. See https://en.wikipedia.org/wiki/Weregild. I think the concepts of feud and weregild did not even exist in the roman world.
But you are right that for a long time you still had the right for revenge, until eventually it was fully phased out. A bit like the transition between the gold standard and fiat currency.
The article related to this post mentions that in Rome, working out monetary compensation for the life of a family member was a common practice. It's not a uniquely germanic-in-origin practice.
Whereas in the barbarian kingdoms and later on, all murders were initially a private matter and people retained the right to kill someone back (not necessarily the killer) unless they opted for the compensation. If this specific case happened after the fall of the empire, probably the father would have just killed the husband back instead of seeking government involvement. And probably the husband's family would have not thought it fair and either seek compensation or kill another member of the wife's family.
Either way, I'm glad things have changed since then!