Many Roman emperors only held power for a few months, and very few held power for over a decade. What did the average Roman think about these constant power transfers? Did an average Roman even know who the emperor/emperors were that year?

While I suppose technically true, popular factoids about how Roman emperors had short life expectancies because of assassination or other causes are based on misleading statistics and a failure to understand what they mean. I touched briefly on this problem in passing relatively recently.

Of the seventeen emperors between Augustus and Commodus--that is, the first two hundred years of the imperial period, and essentially the entirety of the period when there was only one emperor at a time--only six reigned for under ten years. Three of those (Galba, Otho, and Vitellius) were defeated imperial claimants during the civil war of 69, one (Titus) died unexpectedly of disease at a young age, and one (Nerva) wasn't so much an emperor as an interrex. Being rather old and without a family, Nerva was probably not expected to live very long and basically reestablished the imperial succession after the death of Domitian. One of the six (Caligula) was assassinated before reaching ten years. This is also without factoring in the practice of co-emperorship, which really began to take off under the Antonines but has its roots in Augustus' extension of tribunician power to Tiberius. It probably wouldn't be wholly inappropriate, therefore, to add to the list Lucius Verus, who ruled as co-emperor with Marcus Aurelius until 169, just missing the ten year mark before dying of disease. I'm not entirely sure what adding Verus to the list accomplishes other than to pad out the statistics though.

So no, in fact during the empire's longest single period and its most stable one, only about a third of emperors died before reaching the ten year mark. If we discount the emperors of 69, as we ought to, then about one-fifth of the first two hundred years' worth of emperors didn't make it to ten years. Two of those three died of natural causes, and only one was assassinated. The statistics get more complicated during the Severan period, and we start to see why statistics alone are misleading. Like the Flavians in 69, the Severans secured the imperial throne after the death of Commodus left no imperial succession and opened up a period of civil war and rival claimants. Beginning in 193 several claimants or imperial pretenders popped up throughout the empire, the last of which wasn't put down by Septimius Severus until 197. At this point many lists, like those found on Wikipedia, will start listing imperial claimants as full emperors. Many will also start listing co-emperors as full emperors in their own right. The Severans regularly practiced co-emperorship, meaning that people like Geta or Diadumenian get counted when they really shouldn't be. The three major Septimian emperors--Septimius Severus, Caracalla, and Severus Alexander--account for all but four years of the Severan period. All three made it to ten years. The Severans had severe issues with succession, which is why you get people like Macrobius and Elagabalus showing up as a short interlude between Caracalla and Geta.

All this is to say that only in periods of crisis did short-reigning emperors tend to reign, except of course when a ruler's reign was cut short suddenly by disease. Since emperors who died early were immediately succeeded, they're overrepresented statistically. Additionally, emperors most frequently died early as a result of civil war or periods of unrest, meaning that their successors weren't likely to survive long either. Consider the third century. More imperial claimants reigned during the fifty or so years of the Third Century Crisis, an extended period of decades of civil war and imperial division following the collapse of the Severan regime, than during the entire first two hundred and fifty years of the imperial period. Why? Because emperors were being proclaimed all over the place, typically several competing at once, with limited control and in the middle of fighting civil wars with their rivals. Wikipedia lists all of these guys as emperors as if there's no difference between them and Augustus, which is misleading in the extreme. Moreover, consider that following Diocletian's reunification of the empire co-emperorship was not only the norm but was vastly expanded. While the tetrarchy didn't long survive Diocletian himself as a system, the practice of having two emperors, one in the east and one in the west, was the norm in the fifth century. Once again, all these guys get listed by Wikipedia, meaning that you appear to have double the number of emperors for the same amount of time, even though things are working normally.

In other words, the only reason why statistically many emperors did not reign for very long is because during periods of serious crisis and civil war you naturally get many different claimants, most of whom don't last very long. Additionally, the emperorship in the Third Century Crisis is a problem in its own right, but in many respects imperial acclamation during this particularly fraught period was often seen as basically a temporary, emergency solution to immediate crises like military disasters along the frontier. Thus the so-called "barracks emperors," military commanders on the frontiers who were acclaimed emperor by their troops or staff to give them the power and authority to exercise supreme command at their particular location and deal with the threat. Such commanders really didn't have a choice in the matter, and were usually removed once the crisis was solved--obviously this is not at all the norm, and badly weights the statistics. In reality, the overwhelming majority of emperors in stable periods, which constitute the overwhelming majority of the imperial period, had long and stable reigns, even those who (like Domitian, for example) were heavily disliked by the imperial elite. When looking at statistics of this kind we need to consider the actual context of what we're looking at, and keep in mind that often these wild periods of what seem like extreme disruption and agitation are, in the lived experience of actual people, tiny little blips of a few months or a year. Even a period like the Third Century Crisis is "only" about fifty years. Which is long, sure, but it's unusual and it's also very uneven. Long periods of the Third Century Crisis were peaceful and relatively stable, and many of the crises were located far from the imperial centers, along the frontiers and in distant parts of the empire that didn't affect most people directly very often at all.