Would paladins enforce this kind of justice or would they work to change the system into something more "civilized"?
What would a Paladin object to in Viking-style law? That there's no tax-supported police force with a jail? That didn't exist in western civilization until modern times. That criminal law is treated as civil law, with an emphasis on restitution rather than punishment? Perhaps some Paladins would rather chop off thieves' hands than have them give the money back, but I don't see the problem.
Our word for "law" comes from Viking society; there's nothing uncivilized about their legal system.
From
The Vikings and the Law:
Did the Vikings have laws?
Oh, yes they did and we know what they were like. The very word LAW in English is a Viking word. The English word for a local law, e.g. laws about where you can park cars etc., is "by-law". The word "by" comes from the Scandinavian word for "town". Similarly a local election is called a "by-election". There are no indications that the 'Danelaw' in England was more lawless than the areas under Anglo-Saxon rule.
What was the Viking Legal System like?
The 'ting' was the Viking word for a legislative assembly and a court. A criminal was brought here to stand trial. The presumed facts of the case were established by a panel (Old Norse "kvidr") of people stating what they THOUGHT was the truth.
A jury of 12, two times twelve or three times twelve, depending on the importance of the case, decided the question of guilt. The 'law-sayer' told the jury what the law said about the crime committed and the accused was either convicted or declared innocent by the jury.
If convicted, the criminal was either fined or declared an out-law. To be an outlaw meant that the criminal had to live out in the wilderness and no one was allowed to help him in any way, and he was free game for his enemies. They were free to do their best to hunt him down and kill him.
'Ull's ring', the sacred ring of the Norse god ULL, is supposed to have been important at trials.
What about Vikings attacking and raiding other people?
There was no law against war with others (Is there today?). Like the Elizabethans in England and others, the Vikings had no law against piracy as long as it was against the "enemy", so raids outside the "law-area" were not illegal!