Is my understanding correct that the Chinese emperor is believed to be the ruler of the world, or was my understanding too literal?
I've also read here that the Han Dynasty was aware of the Roman Empire and vice versa, but I was also wondering about other neighboring realms (i.e. Japan, Korea, Indochina region, Malayan peninsula). Did they believe them to be pretenders?
Also, if possible, how would a Chinese peasant react upon hearing that there are other empires beside their own?
Your understanding is exactly right. Even until the late 1800s the majority of the Chinese literati felt outraged as the notion that their Emperor is addressed as an equal by foreign entities. I will give you a few examples:
The rivalry of China and Japan could be traced to the point where Japanese rulers were proclaimed "Heavenly Ruler", an epithet which dwarfed the Chinese "Son of Heaven". From this alone you can see how seriously this title was treated with in the East Asian traditional culture.
A tutor of Empress Tongzhi of Qing, Woren, wrote, upon hearing that foreigners were to be made language teachers in the Tongwen College, actively believed that it was improper that ANY foreigner should be made a teacher, the most respected career in Chinese tradition.
Even until the late 1800s, Western nations were still seen as barbaric by the majority of Chinese literati. Westerners were called "dirty animals" because it was believed they had reeking body odours. Most of the Chinese simply did not believe they were "civilized", and would rather believe the military technology they demonstrated to be brutal and monstrous (linked with the nomadic empires that frequently defeated China's armies in the past) rather than a thing of intellectual and technological advancement.
As it could be seen from above, foreign nations were despised and the Chinese, from the Emperor to the people, believed they must be inferior. HOWEVER, the traditional, Confucian monarchical image that Chinese Emperors strived to become was "kind and generous" toward its weaker vassal states. So, if the foreign nation in question was weak and subservient, the attitude of "they are monstrous barbarians" became "they are helpless and our magnanimous selves should offer them assistance". This attitude was literally overnight for the Da Wan, for example. The hardlined, no mercy approach the Hans went with Da Wan occurred ONLY because they would not offer the Han Emperor a really fine horse. Tens of thousands of Han troops died as a result, and once the Da Wan nobility surrendered the kingdom, the Han attitude instantly changed to clemency and they left with a herd of horses, without even so far as entering the city of Da Wan and imposed nothing more to end the war. Another example would be Yang Guang. Yang Guang, Emperor of Sui, believed so much that he must shower his vassals with kindness to show how perfect of a monarch he was, that whenever he held his annual celebration with foreign diplomats, he would offer thousands and thousands of pounds of gold to his vassals. Upon accepting the allegiance of some Turkic hordes, he even went with Turkic tradition to eat in their Khans' tent, allowing them to crown him as the "Sky Khan". Similarly, clemency was pursued whenever China was at war with a foreign power that presented itself as weak.
5）Kublai Khan may not be "Chinese", but his pursuit to be the best of the "Son of Heaven" and his stubborn belief that the entire East Asian region must submit to China lead to some really stupid decisions, such as declaring war on Southeastern nations as far away as Malaysia just because they would not give their tribute on time.
Sources:
Kim, Key-Hiuk The Last Phase of the East Asian World Order: Korea, Japan, and the Chinese Empire, 1860-1882
Chang, Jung Empress Dowager Cixi: The Concubine Who Launched Modern China
Sui Shi
Yuan Shi