The simple answer is no Mongolia is an independent democratic country located in north and central Asia. But there are regions in China and Russia that are home to many Mongolian people whom all talk Mongolian and have close beliefs to that of the Mongolian homeland to the point where you may find yourself confused between them. The region or province that people most commonly get confused about is in China and is known as inner Mongolia. This is the home to most of the Mongolian descendants outside of Mongolia. One of the most interesting differences between modern-day Mongolia and inner Mongolia is the written language. In Mongolia, they stopped using Mongol script many years ago and opted for Cyrillic writing instead whilst in inner Mongolia, they still use Mongol script, and it can be seen used in many places such as road signs and banknotes.
As previously mentioned, inner Mongolia or officially the Inner Mongolia Autonomous Region is a part of China and is governed by them. Inner Mongolia’s total land area is approximately 1,200,000 km2 (463,000 mi2) making it the 3rd largest Chinese region home to around 25 million people as of September 2022. Of the total population in inner Mongolia, only 17% are ethnic Mongolians and they live in mostly rural areas. The ethnic group with the majority is the Han Chinese making up 79% of the population in the region. The official languages in the region are Mongolian and mandarin, other languages used in the region are Oirat, Buryat, Dagur, Evenki, and Jin.
To answer this question depends entirely on how you define Mongolia and China. In the past, during the tang dynasty (618-705 AD), Chinese emperors did control the land known as Mongolia today, but at that time the Mongolian ethnic group wasn’t around yet so it could be argued that the Chinese didn’t control Mongolia as it wasn’t formed yet. But There was a 200-year period during the Qing dynasty (1644-1912) when Mongolia was ruled by the Qing. But the Qing dynasty was controlled by the Manchu people who are not the same as Han Chinese people and are rather a completely different ethnicity. So yet again it could be argued both ways, but the Qing dynasty was a Chinese dynasty so yes at one point the Chinese did control Mongolia.
No, although it at one point was a satellite state of the Soviet Union. The Soviet Union was the main player in helping the Mongolians gain independence from Qing rule in 1911 and helped make it officially independent on the 5th of November 1921 when the USSR recognized outer Mongolia as an independent country. The communist Russians still held a lot of influence in the country until 1991 when Mongolia peacefully transitioned from a communist state to a democratic nation.
During the 13th and 14th centuries, Mongolia went from seven nomadic tribes to being the largest continuous land empire in history. The seven nomadic tribes were brought together under the rule of Genghis khan when leading members of the seven tribes proclaimed him as the leader of the unified Mongol empire. Due to being used to living tough lives and being taught how to ride horses and shoot bows from a young age, it made the Mongolians formidable warriors from a young age. With the knowledge of Genghis khan and the very mobile warriors used to moving around in harsh weather conditions, it made Mongolians an unstoppable force. With all this, the Mongolians managed to control at the peak of its empire around 33 million km² (12.75 million mi²) making it 3 million km² (1.15 million mi²) bigger than the size of the whole African continent (30.37 million km² (11.73 million mi²)).
Mongolia’s current relationship with China is a friendly one that benefits Mongolia because 90 percent of Mongolian trade is with China. To make Mongolia’s relationship with China official a treaty of friendship was signed in 1994 by Chinese Premier Li Peng and since then has remained friendly.
Mongolia’s relationship with Russia is also a good one with 90% of Mongolians during a public survey in 2017 saying they have a favorable relationship with Russia. The Public’s opinion may have changed since the Russia/Ukraine war started but the Mongolian government has remained neutral on the whole situation.