Hi there, first things first. Easy answer first: Yes the Byzantines had slaves, roughly 10% of the population were enslaved and there were slave markets in Constantinople and many prisoners of war exchanged between neighbouring states. Byzantine slaves don't feature much in our sources for daily life in agricultural regions nor in urban centres. One source for a senatorial family mentions 30 slaves for one palace but this isn't generalisable for the empire at large. The Book of the Eparch (10th Century) is a guide to the guilds and manufacturing centres of the city, and it mentions slaves being active in several industries (dyers, tailors, butchers and drovers etc.) but the lexicography for slaves changes a lot. In Ancient Attic Greek a 'doulos' is a slave, by Byzantine times it means 'servant' or 'subject' and the emperor is a 'doulos tou christou, servant of Christ' on many coins so we have to be careful about who is enslaved here.

The Byzantine Empire lasted for over a thousand years so no one answer is really going to answer a question on law. These matters take a thousand pages. By the later period for instance the Orthodox Church handled most criminal trials and relieved the stress on the imperial bureaucracy but for the majority of the empire's life-span there was a clear system of laws and trials. The Codex of Justinian made a clear series of laws in numerous volumes, the ecloga 'selection' of the eighth century made one small law book to streamline things for a difficult era of invasions but always there were judges 'krites,' one author Michael Attaleiates being a judge for his main career and a historian on the side. Such careers required extensive educations and training and were predominantly a product of the capital but were open to intelligent and ambitious provincials.

You also mention the right to a fair trial. This can divide our historian. Kaldellis in the Byzantine Republic would state that all citizens of the empire were beneath the law and that the emperor was not above it but executive orders happened. Different emperors had different levels of authority and Justinian the Great (6th century) as well as Manuel I Komnenos both set out that an emperor's authority came from God and could act as moderator on church and secular law.

At the least, punishments were different to Western Europe. Most aimed to leave the criminal alive and would remove the offending body part. A thief would lose a hand, an adulterous woman her beauty (her nose would be slit), a rapist would be castrated etc. Only murder and the violation of church members and property (which was an offence to God) would be killed and even then just simply by the sword. Not excessively. Where it gets grisly is people who challenged the emperor and these individuals could be blinded or castrated depending on their level of influence. It wasn't always the end at least. Michael VII Doukas deposed in 1078 was castrated and still ended up a decade later as the bishop of Ephesos, by that point, the most important ecclesiastical seat in Asia Minor.

Any further questions on particular periods, just say.