Would they be aware of how historians could exaggerate or even fabricate their histories?

Absolutely. Check out Plutarch's essay, "On the Malice of Herodotus" for one example (though Herodotus comes off rather better than Plutarch in that essay, IMHO).

But just because someone could consider a text critically didn't mean they would. There was a definite bias in favor of eyewitness/at-the-time reports, and, often, a lack of information. Caesar said he faced 250,000 Gauls? Fine. On what basis would one critique that number? What other sources would a Roman of Aurelius' time have for the population of Gaul in 52 BC? Archaeology was not a science at the time. Critiques of a historian's work were more likely to be based on the reputation and possible bias of the historian than anything else, and the further back in history the events being discussed were, the less basis one would have to critique them (see: Plutarch's essay). In general, I think, historians were treated as accurate reporters of the events they witnessed and lived through, and accurate with regard to prior events to the extent that they drew on historians who were living witnesses of those events.

Ronald Mellor explains what history and the study of the past meant to the Romans:

The Roman's deep devotion to their past is evident in their literature, their political and legal instituions, their religion and their funeral celebrations. They were proud of their traditions-what had begun as family memories became over the centuries a national mystique. Cicero provides an important reminder that for the Romans the past was a validation of their present greatness: it had to be preserved to give meaning to the present. Thus the heroic legends of Rome's ancient past were transformed from oral traditions into epic poems, and later the same stories, such as the exploits of Romulus, became the material for historical prose.

So, to answer you question about Caesar, Mellor writes:

To the Roman mind Caesar's commentarii were not histories. There are no reflective prefaces or background information to set the work in its historical context, nor is there any clear moral purpose.

But I think in general, an average Roman would have believed that events in what they considered to be histories were true. But history meant something fundamentally different to the Romans than it means to us. They considered the literary composition of history to be far more important than research. Roman historians did not consider research to be nearly as important as Greek historians like Herodotus, Thucydides and Polybius did. The Romans believed that they shouldn't tell lies in history, but they certainly embellished the truth. Often times, Roman history was written by senators and the like who were essentially trying to validate what Rome was doing in the present. Also, we know more about the early Roman republic today, than they did in the past. The Romans just didn't have a way to research the past, especially anything before 300BC, therefore they could not check the validity of what the historians told them. Even if they did have the ability, attempting to do so would essentially be looked at attempting to discredit Rome. Roman history was first and foremost a way to make moral judgements. Essentially, Roman history was the study of morals in the same way that Greek Philosophy studied morals.

Mellor goes on to say that because Roman history tended to be written by Senators, for Senators, it tended to narrowly focus on political and military events. Thus

School teachers and the general public might look elsewhere for stories of the past: poetry, biographies, and collections of anecdotes. But if the general reading public was less interested in history than in poetry, senatorial historians cared little since their work was an extension of political life and was aimed at those who held political power in Rome.

So in short, Roman history was tightly focused on the moral teachings and less so on a strict adherence to events exactly as they happened.

Ronald Mellor. The Historians of Ancient Rome: an Anthology of the Major Writings.