Give your home guard and hammerer wooden crossbows they always carry, they give out beatings with those and it's pretty weaksauce, helps keep the doctors happy and hardly any dorfs suffocate from broken necks.
Anyway.
Conviction is always manual, keep you busy during a tantrum spiral, back up to seven pages of recent convictions in my current fort.
No idea how sentencing works, but dorfs dislike being caged more than being chained, and you can give them a little happiness when chained with surrounding furniture and stuff. Have lots and lots and lots of chains, suggested amount of 1/4 dorfs probably won't fill up.
Something that can happen, is if the sole witness is a good liar, he might be the actual criminal, vampires will try it out, but often dorfs commit multiple offenses one after the other, and it's just the same guy on a tantrum, so normally don't worry about it. Sherrif will never report himself, so if you have lots of crimes with no reports, it's totally the Sherrif.
miscarriage of justice, I hear gives everyone in the fort a small bad thought, but I am yet to hit it, they're not that sneaky, I can just about always pick the crim from no witnesses, there are combat reports after all.
The penalty for no justice is the guy who got beat up or whatever has quite a lot of bad thoughts because of getting beat up, and justice counter-acts that for them and gives them and all their family and friends get a strong good thought. It's a very big deal to get a murderer convicted because their whole family and possibly chains of friends may go insane if you don't, as one going tends to take the rest.
Reports are in order of the date the crime was committed, top are oldest, and adjacent are often the same dorf.
Cold cases are just really old crimes.