Alright. In response to your dump of questions I will give back an information dump. I will keep it relatively brief and will leave it up to follow up questions if you want to know more.

Firstly, China does not have a common language and never did have a common language. As any linguist would tell you, written language =/= spoken language. Classical/Literary Chinese is a writing system devised around the spoken language of Old Chinese, the ancestor of all modern Chinese languages. Asking why people spoke Old Chinese is like asking why people spoke proto-Germanic or something; that is, no one chose anything. But Classical Chinese, the writing system, was the primary way literature was written until basically the 20th century and it is not the same as the Chinese languages that people were speaking or are speaking now. Essentially, the written Chinese was a different language than the ones people would be speaking in disparate regions. Classical (written) Chinese became the standard of writing under early dynastic China and spread as dynasties occupied territory; this was the only writing system in East Asia at the time (SE Asia had its own writing systems though). Nowadays, Chinese writing is vernacular; that is, it is written based on the given modern spoken language, be it Mandarin or Cantonese or Hokkien, or any other.

I will answer your question about Viet Nam and your question about the rest of SE Asia together. VN is an interesting country in Southeast Asia in large part because of its atypical outside influences. I say atypical because the rest of SE Asia (Thailand, Cambodia, Myanmar, Indonesia, Philippines, etc.) took a lot of their "foreign" borrowings from India (or what would become India after the colonial period anyways). In contrast, VN's biggest borrowings into its traditional culture were from Chinese culture. Along with this, Viet Nam borrowed Classical Chinese writing as well as developed its own writing system based off of Chinese (similar in some ways to what Japan did). It should be noted that Vietnamese language itself is not related to Chinese. It is a Mon-Khmer language, so it is related particularly notably to Khmer, the language of Cambodia. However, VN was also influenced in part by India via its relationships with Champa and Cambodia to the south. VN effectively became a bit of a melting pot of Asian cultures and traditions (particularly in the central and southern regions) from the 1500s onward. Going off of this, I will say now that with the exception of Viet Nam, all of the writing systems of Southeast Asia are related. From the Philippines to Thailand to Cambodia we can see that the writing systems are ultimately Brahmic scripts, which are derived from Indian writing lineages.

u/KippyPowers is mostly correct but unfortunately misguided on a few points, feel free to challenge me on anything you think is wrong.

1. Sino-Xenic languages like Korean, Japanese and Vietnamese spread primarily through literature. Most of these had no official script prior to Chinese and would essentially have adopted the first one they encountered. As it is, the nascent states encountered China very quickly. Japan had official relations with the State of Cao Wei in the Three Kingdoms period before Japan was even unified. Vietnam was essentially ruled by Chinese rulers since 200BC, and Korea had extensive contact since even before that. The real proliferation of Chinese writing was actually due to the pressing need for literacy to emulate the Chinese court bureaucracy but also due to the spread of Buddhism and Confucianism around the Tang Dynasty.

2. There is an official Chinese language and there has always been one, though it has not been the same one. While there have been an almost uncountable number of dialects, Chinese bureaucracy always ensured that there was an official one which was spoken for official business and as a lingua franca in courts and debates. Today Chinese is based mostly on the Beijing dialect where the capital is, but until the mid-Ming Dynasty was based on the Nanjing dialect where the capital was, which are not mutually intelligible. We actually have a pretty good idea of what that sounded like because until recently Chinese dictionaries were rime dictionaries, sorted by official pronunciation, where today they are sorted by brush strokes. The most famous of these is the Qieyun written in the Tang Dynasty.

3. China has tried multiple times to conquer Korea, and succeeded a few times. The first was in the Han Dynasty, which established the Four Commandaries in Northern Korea. At the time South Korea was barren and the Koreans were an ununified tribe, so there wasn't too much point expanding further South. The Mongols in the Yuan Dynasty succeeded several centuries later, integrating Korea into a loosely unified China. By the Ming Dynasty, China was more interested in keeping soft power control over Korea. Two key reasons exist for why China never actually conquered Korea permanently. Firstly, the Changbai mountain range and Yalu River essentially made access into Korea impossible except through the choke point of the fortress city of Kanggye. Even after bypassing that fortress, 80% of North Korea is rolling mountain ranges and deep valleys, making the terrain a strategic and logistical nightmare. Secondly, Chinese empires have not typically been concerned with expansion, but rather prosperity. Korea was almost never an existential threat unlike the nomadic steppe hordes, and offered little trading opportunity unlike Vietnam, and so was of little interest, especially since it was overwhelmingly obedient.

4. The simple answer is that Vietnam was conquered by China for most of its history. The entirety of South Vietnam was actually a different political entity called Champa, which was Hindu and very Indian-influenced. North Vietnam had been conquered in 111BC from a Chinese warlord, and would remain under Chinese direct control for the next 1000 years. Burmese and Thai writing were both descended from the Pallava script from South India. What is often forgotten is that the entire region was quite staunchly Hindu before it became Buddhist (which is really a branch of Hinduism anyway), monks and priests quite naturally adopted these. Thai is actually very Sino-Xenic, with an estimated 40% of vocabulary having Chinese origins (including all the numbers except 1 and 2).

5. Southeast Asian writing systems are actually very similar to each other and mostly have the same roots. Thai, Cambodian, Laotian and Burmese are all descended from the Pallava script from South India. The Pallava script was also initially used throughout Malaya, but was displaced by the Jawi Arabic script with the advent of Islam. Vietnam used Chinese. Under colonial influence, Vietnam and Malaya switched to the Latin alphabet.

6. Absolutely. The literal and cultural heritage of the Mongols is linked with some regions that would seem almost improbable. The Mongolian script is descended from the Old Uyghur script (today the Yugurs, the modern Uyghurs are not the Old Uyghurs), which is descended from Aramaic (aka the language of Israel, Jesus and the Bible), ultimately descended from Phoenician. The Greek alphabet is also descended from Phoenician, and Latin is descended from Greek, which means the Monglian script is directly related to the Latin script. Culturally, the steppe nomads can almost all trace some kind of heritage to the Xiongnu. This was not necessarily a linguistic one. Mongolic, Turkic, Tungusic, Tocharian and Uralic tribes mingled constantly with each other in the steppes, resulting in a mash of cultures and religions. Genghis Khan's coronation at the Baljuna Covenant in 1200 had representatives including Nestorian Christian bishops, Muslim imams, Buddhist abbots, legendary Taoist sage Qiu Chuji, and of course Tengri shamans. In fact, the Khazar Khaganate was Jewish.