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I will break this down into sections. Not an expert but I did take a college course on Rome. This question is a little broad so I will go with the Roman Republic era and stick to citizens of Rome for this purpose. It would be a much bigger answer to cover then varying flows of legal traditions throughout the near thousand year history of Rome, before the fall of the western Roman Empire. Even more if the continued traditions of the Eastern Roman Empire were included.
Early Roman laws were based on the Twelve Tables, which was a simple set of laws that the rest of the Roman legal system was based on.(Twelve Tables, n.d.) It is also important to understand the two citizen classes in Rome, the patricians and the plebeians. The patricians being a ruling class of people, who connected through a network of favors were at the top of the economic and political food chain. Basically the leaders of important families and clans with in the Roman society, in early Roman society the defining lines were blurry between cast, but is was solidified by King Severius Tullius, the first Roman leader to begin recording and ranking roman citizens. (Editors of Encylopedia Britanica, n.d.) This concept is important since it sets the tone for the entire Roman Republic and later on the Empire.
Legis Actiones is the first format and the more common version during the Roman Republic time period.
So a case began as a complaint to the magistrate. "Severius broke my barn door and let out my sheep in a drunken fit last night, he owed me x amount for reparations" said Tiberius. The initials complaints, preliminary hearing. A decision was made at this stage, appeals were possible. It was very political and complex. Read this if interested (http://romanlegaltradition.org/contents/2013/RLT9-METZGER.PDF) The author of this outline wrote an entire book in the subject. It requires a paid account to an academic site that I do not have access too. It is not cheap to buy since its an academic text book as well. This question is an entire history and law orientated college class.
Long story short though, it appears to be the defendant and the prosecutor came to some sort of agreement and were guided by the magistrate during the pretrial arrangement. Depending on the time frame this was with a Consul, a Military Tribune or a when the office of Preator was created it was the mostly the Preator.(Wikipedia, n.d.)
Summons
Summons under the legis actiones system were conducted in ius vocate, that is, >by voice. The plaintiff would request, with reasons, that the defendant come to >court. If he failed to appear, the plaintiff could call reasons and have him >dragged to court. If the defendant could not be brought to court, he would be >regarded as indefensus, and the plaintiff could, with the authorization of the praetor, seize his property. The defendant may elect a representative to appear >in his place, or seek a vadimonium - a promise to appear on a certain day with >a threat of pecuniary penalty if he failed to appear (Wikipedia, n.d.)
Once all this was taken care of it went to a Judge. Which was usually of a lesser rank and determined by the two parties.
Preliminary hearing At the first stage of the case, a hearing took place before the praetor, in order to agree the issue and appoint a judge. This was conducted through exchanges of ritual words, the two different types being known as the declarative which were the legis actio sacramento (which could be in rem or in personam), legis actio per iudicis arbitrive postulationem and legis actio per condictionem and the executive type legis actio per pignoris capionem and legis actio per manus iniectionem.[2] All of these involved, essentially, statements of claim by both parties, and the laying down of a wager by the plaintiff. Then, a judge was appointed who was agreeable to both parties, the praetor making a decision in the event of a disagreement. Judges were chosen from a list called the album iudicum, consisting of senators, and in the later Republic, men of equestrian rank. (Wikipedia, n.d.)
So pre-trial politics and maneuvering are complete at this stage, the two parties have agreed to what the actual trial will be, the praetor or magistrate has made decisions and Judges are selected from the appropriate group of people.
Full trial
Once the judge had been appointed, the full trial could begin. This was fairly informal compared to the preliminary hearing, and was supposed under the Twelve Tables to take place in public (the Forum Romanum was frequently used). While the witnesses could not be subpoenaed, the dishonorable status of intestabilis would be conferred on a witness who refused to appear. There were few rules of evidence (and both oral and written evidence was permitted, although the former was preferred), although the burden of proof lay upon the plaintiff. The trial consisted of alternating speeches by the two advocates, after which the judge gave his decision. (Wikipedia, n.d.)
Finally after having been through all this process and a verdict is given, it was the responsibility of the victorious party to enforce the verdict.
Execution Unlike in the modern legal system, victorious parties had to enforce the verdict of the court themselves. However, they were entitled to seize the debtor, and imprison him until he repaid the debt. After sixty days of imprisonment, the creditor was entitled to dismember the debtor or sell him into slavery, although after the Lex Poetelia of 326 BC, the creditor could take no action other than continued imprisonment of the debtor. (Wikipedia, n.d.)
My interpretation of this is that you did not go to trial for frivolous reasons, since it was a serious affair and took quite a bit of time and political maneuvering. It was a long process full of faults and resolved in limited action. It was later replaced with other systems that are outside of the scope of the time frame I am narrowing down too. F
That just about covers all the questions from the perspective of the Roman Republic period, early to pre-Empire. It is a bit rough, but this is a well studied academically. So to highlight the questions in brief for clarity.
How was it decided if a case should be tried in Rome or in the province by the governor of that province? At least during the Republic period a case was determined by the proper authority and a pre-trial event took place were certain criteria were met.
Was the governor always the judge or were there exceptions? The leaders of the province were involved with hearing the initial issues and getting things set into place, but were not the judges, at least not in the above quoted system. Though the Preator could be a judge if both parties agreed to it, this did not have to be the case.
who were the lawyers? The plaintiff and the defendant were the lawyers. You were expected to know the laws if you were a Roman citizen. Since the laws did not stack up and cause a legal tangle like our modern system, it was not as hard as it is now. Though it would of still required some effort. Before a citizen would of went to trial the legalities would of already been debated out as well.
I know a lot of senators were lawyers but what about outside the city? Senators were all over the place. They were any high ranking official that was still alive. It was not something that you got elected into technically, it was a title for some one who has achieved a certain political rank. Roman political culture was quite different from what we are used to. It was frowned upon to hold office for long periods of time for instance, usually more then a year or two. Once some one reached a certain rank in the political system they remained a Senator and had certain duties to the Republic.
Were there layers in every province that were defending cases against the governor or was it that you should represent yourself?
People represented themselves. Citizenship was viewed a bit differently during this time. You were responsible for a lot and citizenship was considered a privilege. One that was compelling enough to get people to join the Roman Armies for 20 years to earn it.
Now things shift into a different set of things in the later Roman Empire. The entire nature of the Rome shifts with the implication of an Emperor. So during the imperial Roman time period this answer is not as complete. Parts of the Roman Republic culture linger on into the Roman Empire, but quite a few things shift. There was also a different system for non-citizens. I am not certain how that worked. I know there was limited protection for the non-citizens. Citizenship was a big deal within that culture.
Editors of Encylopedia Britanica. (n.d.). Patricians. Retrieved from Encylopedia Britanica: http://www.britannica.com/topic/patrician
Metzger, E. (2013). An Outline of Roman Legal Procedure. Retrieved from Roman Legal Tradition: http://romanlegaltradition.org/contents/2013/RLT9-METZGER.PDF
Twelve Tables. (n.d.). Retrieved from Illustrated History of the Roman Empire: http://www.roman-empire.net/republic/twelve-tables.html
Wikipedia. (n.d.). Roman Litigation. Retrieved from Wikipedia: https://en.wikipedia.org/wiki/Roman_litigation