Well, I can't really speak to the Ming or Qing dynasties, but I can speak to the Han dynasty. It's far before the Ming/Qing, but it might be interesting.

For starters, a bit about regnal names: when they took the throne, Chinese emperors effectively did away with their names. It became a rather huge faux pas to use the emperor's personal name in pretty much any form. This proscription went so far that people would change their names to avoid using the emperor's name.

It would, of course, be impossible to observe the imperial taboos unless you actually knew what was taboo. Which, in turn, required knowing the emperor's name before he ascended to the Dragon Throne.

When the emperor died, he was almost always given a posthumous name. The first emperor of a given dynasty might retroactively give their parent an imperial title, regardless of what post that parent held (Sun Quan of Wu somewhat famously gave his father, Sun Jian, an imperial title... about four decades after Jian's death). These posthumous titles always meant something. Qin Shihuangdi meant "First Emperor of Qin." "Wendi" meant "cultivated (cultured) emperor." "Andi" meant "Peaceful Emperor." The di character was shorthand for huangdi (emperor); the long-form is sometimes used instead (nobody's really going to flip tables if you refer to Liu Bang as Gao Huangdi instead of Gaodi... although that's not a great example because he's often known as Gaozu instead). Most of these posthumous names get the dynastic name added to the front for the sake of convenience. And the dynastic name can be the name of the family or of the state; Shu-Han's dynastic name is Liu (after the family), whereas the Han's dynastic name was Han (the family was still Liu).

Additionally, emperors had regnal years. These years (which always began with the Chinese New Year are typically how you're going to see Chinese historians refer to events. The death of Liu Hong in 188 might be recorded as: "in the fifth year of Zhong Ping (Central Stability), Han Lingdi died." Zhong Ping, by the way, actually spanned three emperors -- Lingdi, Liu Bian (whose reign was so short he doesn't even get a posthumous name), and Xiandi. Emperor Xian proclaimed Chu Ping (Beginning Peace) on the first Chinese New Year after he ascended to the Dragon Throne.

You might also have noticed the extremely optimistic nature of regnal years. They're frequently named things like Radiant Harmony (Guang He), Commence the Revival (Jian Xing), and so forth. The average peasant would almost certainly have been familiar with these reign years; Sanguo Yanyi mentions how the reign year was often proclaimed or announced. It's more historical fiction than a proper secondary source, mind, but it's usually pretty accurate on the cultural notes like that.

But, for the most part, the emperor didn't factor into the daily lives of the peasants. The peasants were far more affected by the local governors and such.

Of course, all bets are off when you start talking about things like the Imperial Examinations. It wasn't terribly uncommon for a peasant (once nominated as "filial and uncorrupt" or xiaolian) to be brought to the imperial capital and, if successful at the examinations, given an imperial posting. If said peasant happened to be given a posting in the imperial capital, the peasant-turned-bureaucrat would absolutely have learned a great deal more about the imperial family and the various ministers (especially if they received a posting directly under a minister).

Of course, it wasn't just peasants who took those exams. Nobles did, too. One of the more famous xiaolian was probably Cao Cao (perhaps the most interesting figure involved in the decline of the Han), who qualified for his first posting under the Han through the exam system.

Not sure about the Qing but during the Han dynasty the people had a saying, "Heaven is really far away". Basically the lives of the ruling class did not really affect their daily lives. They knew of the Emperor and his rule, but most of the problems and conflicts were handled by the local authorities. The Emperor and his family was also pretty secluded from everyone else because they lived in the Forbidden City so the average person wouldn't know about things like who the Emperor's wives or court members were.

Source: "Daughter of Han- an autobiography of a Chinese working women"