It's a stupid hour for me to try to be making sense, but I'm gonna try because I think you're great
Since I'm the government guru 'round here, I'm going to try to get a lot of this stuff settled for ya.
1) There's a dungeon beneath the castle, but that's not where the petty thieves and such end up. The big-time criminals are held there along with those who have committed crimes within or directly offending those within the castle. Your everyday naughty guy would end up in a jail by the barracks (as the military handles the patrols of the city and most law enforcement stuff- they're Escova's cops). This would probably be a collection of cells beneath a building A similar jail would exist in Kyha. You're right in thinking they aren't really meant for long-term incarceration. I imagine there would be sentences involving a loss of property, a lot of fines, forced labor, exile, possibly death, etc.-- things that get people out of the jails and either back into society to be productive or out of everyone's hair.
2) Escova's largely comparable to a Feudal judiciary system in that the nobility is responsible for acting as judges outside the cities. For example, if a crime happens on a baron's land, he's responsible for hearing the case and determining guilt and punishment. A baron accused would be tried by the court of his duke. The nobles would be assisted by their vassals (knights, lower nobility, etc.), who would be called to help in the administration of justice (much like the King and his court). Appeals would be made to the King, if necessary.
Within cities, criminals are handled by the military. In the case of Maristheum, the high-ranking officers are tasked with the judicial process (captains, the High General . . .). Particularly severe offenses would be sent to the castle to be heard by the King and Queen.
The military handles crime within its own ranks, and the Mage's Guild is responsible for the punishments for magical crimes.
3) There isn't a lot in terms of written law. Much is decided based on a noble's judgment, precedents, and the like. Assume that the punishment for a lot of the common crimes is the payment of fines. Punishment may also be determined by the offender's personal means. After all, you can't say that the punishment for theft of X thing is to take a man's goat if not every guilty person even owns a goat!
Also, refugees can definitely run into trouble for administering their own forms of justice. The military may not always find out about that, but it's not permitted.
4) The penalty of death exists, particularly for repeat offenders of the more serious crimes, treasonous persons, etc. Because there isn't some massive code of law with predetermined punishments, I can't quite provide a chart of common offenses and their repercussions, but you could expect things to be a little on the harsh side but without a lot of "Well, you did something naughty, so stay in jail." The Escovans don't do a lot of sheltering and feeding criminals for a long time just to keep them from stealing or doing naughty drugs again. They're not quite so nice.