This may not give you a direct answer, but one of the most important tools emperors had in shaping juridical policy and law was rescripts (rescripta), responses to petitions for the emperor's input on legal matters.

These responses were addressed typically to public officials who had sent letters to the emperor asking for help in resolving a legal dispute or in implementing official policy where the law was vague or did not apply as written. The most famous set of such petitions came from Pliny the Younger and were addressed to the emperor Trajan. Many had to do with the subject of Christians and how Pliny should deal with them as governor of Bithynia and Pontus.

These requests came, at some point, to be called libelli ("little books"). And it appears that emperors became increasingly pressed for time in dealing with them (quite probably because of how labor intensive it became to be emperor and to not die during the third century). They began to employ officials whose primary job was to respond to the petitions, and those officials were called magistri libellorum ("masters of libelli"). I should mention here that ALL emperors' rulings held legal force in ALL jurisdictions unless they were countermanded directly by subsequent emperors. This would cause untold headaches, which I'll address briefly in a minute.

Crucially, all this does not mean that local governors or public officials would consult the emperor about every jurisprudential decision they would be making. The Romans did love delegating responsibilities to local officials, even after the imperial government brought a greater degree of standardization and logic than had existed during the Republic. So how important these magistri libellorum's jobs were in the grand scheme of imperial governance is difficult to say succinctly, but it's worth pointing out that two of them, a Gregorius (or maybe Gregorianus, though probably not) and Aurelius Hermogenianus took up the task of compiling, editing (?), and publishing collections of imperial legal decisions and proclamations from approximately the 130s through 294. (The complete documents do not, afaik, survive, so the Gregorian codex, which covered the pre-Diocletianic and early Diocletianic period, may have stretched back earlier.)

Why was this important? Well, aside from forming a core source for Justinian's Corpus Iuris Civilis, which still informs modern legal thought and doctrine, it indicates that the emperor's word had become something rather difficult to ascertain in legal matters. A particularly adept lawyer in Lusitania in the 280s representing a plaintiff litigating over a land deed could counter an argument based on the previous emperor Carinus's opinion on a case in Sirmium by citing an obscure ruling given by Macrinus over a similar question about land near Carthage decades before. (This is a hypothetical example.) Both opinions had legal weight, even if they were contradictory.

So to circle back to your questions:

What was the emperor's role in the judicial process? The emperor could make legal rulings as a magistrate, but that probably wasn't among the most pressing (or interesting) of his duties. Tiberius apparently rather liked doing this. The emperor also could "advise" local officials on how to handle specific cases or matters of legal policy. I suspect that this wasn't all that important to most emperors personally, but I also think that over time, the lack of codification and the concomitant workload in sorting out ambiguities and contradictions probably started to drive emperors a bit crazy. Gregori(an)us and Hermogenianus may have been the first individuals tasked with fixing this, but they absolutely were not the last. (cf. the efforts under Theodosius II, Alaricus, Justinian, et al.)

Is Paul's story rooted in [real and practical] aspects of the judicial system? I . . . don't know. The historicity of ANYTHING in the Bible should be treated with great circumspection unless it's verified by (unabashedly frankly) better sources. With that said, Paul is claimed to have been a Roman citizen. This is important because citizenship was, as far as we can tell, very rare in distant provinces except among high-level provincial and local administrators and the inhabitants of coloniae during the period. Assuming maximal historicity (LOL), Paul may well have had the ear of one of the emperor's delegates. The passage from which you drew the quotation certainly indicates that Paul thought it within his rights to call on the emperor for adjudication of his case. But then again, the emperor may just as well have left it in the hands of local officials (magistrates and citizens), and no higher authority would prosecute the local magistrate for failing to allow Paul this opportunity unless Paul were of personal significance to such higher authorities. The Roman Empire did not have a developed system of public prosecution.