pretty simple, during the roman empire, around 200 bc when there was lots of political instibilty. did plebeians have the power to vote, was it still a democratic republic
Rome never was a democracy. Yes there were citizen's assemblies and citizens could vote in those, but it wasn't the "one person, one vote" system we know these days. Instead, citizens were organised in tribes, centuries, or curia (depending on the assembly). Citizens could vote how their tribe/century/curia would vote in the decision. These tribes/centuries/curia would be stacked against the common people. Let me discuss the assemblies and explain how the common people got the short end of the stick in each.
Popular Assemblies
The Curiate Assembly is the oldest assembly of ancient Rome, but many of its powers were transferred to the other assemblies at some point. For most of Rome's history, the Curiate Assembly validated the powers of magistrates after they were elected. That is, other assemblies would elect magistrates but the Curiate Assembly would 'approve' them by giving the elected magistrates their formal powers. The Curiate Assembly gave "military auspices" to consuls, praetors, dictators, and anyone with imperium (occasionally a person who was not elected as a magistrate would be given imperium by a popular assembly for a specific task).
Formally all citizens were part of the Curiate Assembly, voting as part of the curia (one of 30) they were part of. Originally the curia were likely wards or neighbourhoods of the city of Rome, but with the expansion citizens from outside Rome were fitted in somehow. In practice this assembly met in skeletal form and rubber-stamped whatever came before it. There was no real power for most citizens here.
The Centuriate Assembly elected consuls, praetors, and censors; it voted on war and peace; it passed laws; and it conducted important trials (particularly if the penalty could be death).
This assembly found its origin in the army. Citizens who were eligible to serve were divided in 188 centuries ranked by wealth. All citizens too poor to serve (eligibility was determined by wealth) made up 5 centuries. Each century had one vote, which means that the richest citizens (who had few members per century) carried a lot more weight than the poorest citizens (who had just five votes between them). Furthermore, the richest centuries voted first which resulted in decisions being locked in well before the poor would cast their vote. Again there was no real power for most citizens here.
The Tribal Assembly also consisted of all citizens, but here they were divided into tribes. The city of Rome had four tribes, the surrounding country had 31 tribes. Citizens voted within their tribe, the majority deciding how the tribe as a whole would vote in the Tribal Assembly. A majority of tribes then decided the outcome of the vote. The assembly voted on legislation and minor magistrates, and held trials.
Here too a large part of the common people was concentrated in just a few of the tribes. The citizens living outside Rome were fewer (for much of Rome's history at least) and generally better off (on average certainly). The poor citizens living in Rome had just 4 of the 35 votes.
There was one assembly where the common people did have lots of influence: The Plebeian Council. Only plebeians were part of the assembly. A plebeian was anyone who was not a patrician, and a patrician was someone who could trace their lineage back to the original elite which deposed the kings (while sticking to certain standards for wealth and marriage).
The Plebs elected tribunes and some aediles. It also passed legislation, which had to be approved by the Centuriate Assembly. Over time the importance of the Plebeian Council grew, with some politicians appealing directly to the Plebs to push their agenda (and other politicians despising such actions).
However, the Plebs really is not a democratic popular assembly, as it excludes a part of the population! Patricians had no vote here. So though this was an assembly where the common people did have huge influence, it's even less democratic than the other assemblies, where at least in theory everyone had a vote.
On political instability
You mention 200 BC as being a time of lots of political instability, but it really wasn't. That period is arguably the period where the Roman system of government worked best. Major conflicts between the patricians and the plebeians had been solved more than a century before (with plebeians gaining access to important political functions) and for a while things went pretty well. It is only later, from the second half of the second century BC onwards, that new conflicts between the classes start arising. This is when you get the controversial agrarian reforms, the military reforms, and the civil wars.
In 200 BC everything went swimmingly. Wars were fought and won, loot came in, there were new lands to settle, there were few politicians trying to grab power unconstitutionally. In short, a perfect time for the Roman Republic.
Sources
This answer is based on a previous answer I have given.
Books and articles: