Why would anyone want Mongolia? The Yak herds? A lifetime supply of Yurts?....
Also, the idea of China coveting Siberia for its lebensraum is naive. China's population is set to decline by 500+ million this century, due to the one child policy they've had for 40+ years. Their focus is on maintaining their borders and integrating Taiwan.
If China wanted to expand beyond its borders, it would have no problem taking over Mongolia, which is a huge country, or maybe splitting Mongolia with Russia. This is a country and people a quarter the size of continental US with a population smaller than that of metro Seattle, and a country with a very long and continious history of invading both China and Russia, so from a historian nationalist perspecitive, it would be a somewhat justifiable payback... Mongolia is completely landlocked between Russia and China, it would be conquered in a weekend by China.
I don't think a Chinese plan to invade Mongolia is in the works, but the point here is that it would be the low-hanging fruit for any expansion plans beyond the Chinese areas of Taiwan/HK.
Hitler wanted living space for Germans to colonize. The Chinese would not be interested in that, but would be interested in natural resources.
Russia has perhaps the top highest total value of natural resources of any country in the world. There is a handful of countries out there that could be completely self sufficient: Russia, the US, Canada and a few others who have oil, minerals, timber, farm land, etc. China is not one of them--they will always be dependent on others, and how is that fitting for the future ruler of the world?
China has an economy 8 times larger than that of Russia and a population 10 times larger.
The enemy of your enemy is your friend and all that, but what do the atheistic wannabe rulers of the world in Beijing really have in common with Moscow other than a dislike for representative government? If China is being friendly with Russia, it is because they want to buy what they have. But why buy it if you can take it? China has no loyalty to anything but the ideal that they will rule the world. Russia may be useful to them now, but tomorrow? Russia has to realize that, which is maybe why they are not selling them 5th generation fighter planes.