One interesting way of communicating throughout the empire was via the images on coins. Fronts of coins would often display a dignified bust of the current (or putative) emperor, while the reverse would show a god or scene of Roman pride, with a slogan inscribed along the rim.
One of the more propagandistic coins was the "barbarian hut" FEL TEMP REPARATIO follis during the time of Constans/Constantius II. It shows a Roman soldier, spear (usually) down, leading a germanic waif out of some wretched cottage in a forest. The inscription means roughly "return of the good times". It was probably meant to propagandize the introduction and settlement of foreigners into the lands of the empire.
Here is a very detailed explication of the coin and the several elements.
Another interesting one was the Hadrian LIBERALITAS sestertius reverse, which shows the Emperor acting as the patron of the people overseeing the distribution of largesse. The inscription's root word is, of course, "liberality". Note the Emperor is sitting on a curule chair, a symbol of power deliberately designed to be uncomfortable. Note the Emperor has no back support and holds himself up with one hand. He is also sitting on it sideways, a concession the artist had to make to capture the scene but still give the chair its distinctive curved-X shape.
No, not in the sense that we think of it. Propaganda is fundamentally only possible with the sort of industrial production techniques available to modern states--a pre-industrial society quite simply lacks the social penetration that a modern state is capable of. This is not to say that the Roman Empire did not have imagery. Certainty, portraiture of, say, Augustus was carefully crafted and filled with political and social significance. The vast majority of these were not contracted by the state, however. Coins were also a way of communicating particular ideas, such as victory in battle or notable justice. But none of these amounted to the sort of flyers and propaganda posters you might see with a WWII era state, Rome simply lacked the reach and productive capability.
The classic text on this is Paul Zanker's The Power of Images in the Age of Augustus.