Why has China had domesticated cattle/sheep/goats for a long time, and while Mongolians utilized dairy in their food we see almost no dairy in Chinese cuisine at all?

In the area covered by modern China, dairy was extensively used in (a) Tibet, (b) the Turkic west of China, (c) Mongolia (Inner Mongolia still being in China), and (d) Manchuria. During the Qing Dynasty, court cuisine featured dairy products. If we exclude the cuisines from these areas, and dairy-using Manchu-influenced court dishes, from being "Chinese", we are left with Chinese cuisines with minimal use of dairy products. Such separation of cultures, cuisines, and peoples in China into "Chinese" (i.e., Han Chinese) and "non-Chinese" is thoroughly traditional, and the use of dairy products was for long a marker of non-Chinese barbarian-ness.

Han Chinese contempt for "barbarian" cultures is sometimes used as an explanation for the lack of adoption of dairy products in Han Chinese cuisine. However, economic explanations suffice. China can be divided into three main agricultural regions:

A rice-farming region, covering the south-east.
A wheat-farming region, covering the north-east.
A pastoral region, covering the west and the far north (including Inner Mongolia and northern Manchuria).

Generally, the pastoral area is the Mongolian, Tibetan, and Turkic areas of China, each with their distinct "non-Chinese" cuisines. In the rice and wheat regions, the two main areas with significant traditional dairy use are Manchuria and Yunnan, cuisines that are also "non-Chinese". The overlaps between agriculture, cuisines, and ethnicities can be seen in these three maps:

Agriculture: https://i.pinimg.com/564x/97/c3/f4/97c3f4a72be7c63a1e45cd8daba3cd14.jpg
Cuisines: https://i.pinimg.com/564x/de/c9/f5/dec9f5cfe0e2b0163183c6a31b5df983.jpg
Ethnicity: https://i.pinimg.com/564x/2d/f5/c7/2df5c7a5f137a1e01f7ca76f2132cadd.jpg

We don't have reliable figures for livestock production until relatively recent times in China. We can look at dairy and livestock production in 1995 (avoiding the huge growth in dairying over the last 2 decades):

Product	Rice region	Wheat region	Pastoral region	Total
Cow milk	1.0	3.2	1.5	5.8
Buffalo milk	2.2	.	.	2.2
Beef	1.2	1.6	0.4	3.2
Mutton	?	?	0.5	1.0
Goat	?	?	0.2	0.9
Pork	.	.	.	36
Chicken	.	.	.	6.5
Eggs	.	.	.	13.4

Quantities are in millions of tons; the 1995 population was about 1.2 billion, and these can be converted to kg per capita per year by multiplying by 0.83. The numbers are from http://www.fao.org/3/x5878e/X5878e04.htm

These correspond to per capita annual consumption of 40kg of meat (30kg of this as pork), 11kg of eggs (about 185 eggs), and dairy products equivalent to 7kg of milk. Note that the majority of the meat and milk comes from the rice and wheat regions, not the pastoral region (which has a very low population compared to the rice and wheat regions). This meat and milk consumption in 1995 is a large increase from previous decades, and this increase is directly related to growth in wealth (similar increases took place in South Korea and Japan). Dairy consumption and production has increased enormously since 1995, and production is currently about 40 million tons of milk, which is supplemented by imports.

Much of the milk drunk in China is in the form of yogurt, notably nailao, 奶酪, AKA Beijing yogurt. Perhaps the best known brand of this type in the West is the Japanese brand "Yakult" (also sold in China for about 2 decades now). Such drinks are usually sold in single-serving containers, usually 65ml in my local supermarkets. The 1995 dairy consumption, about 19ml per day, would be about 2 servings of such yogurt drink per week. Since some of the dairy is consumed as cheese (especially in the Turkic and Tibetan regions and Yunnan - Yunnan features Tibetan styles of cheeses among local Tibetan peoples, and cow and goat cheeses such as rushan (dried sheets of cow cheese) and rubing (a halloumi-like goat cheese, frequently fried)), milk/yogurt consumption would have been lower than this.

Noting that this small dairy consumption - 19ml per day - is already greater than traditional consumption in, e.g., 1900, the lack of dairy in Han Chinese cuisine is unsurprising. Dairy is not absent in Han Chinese cuisines. Traditional dishes that use dairy include partly-dried milk skin wrapped around fillings to make sweet "dumplings", fried milk, and milk tea.

Finally, a short note on lactose intolerance. Han Chinese avoidance of dairy is sometimes attributed to lactose intolerance. However, the dairy-used non-Han populations in China also have very high levels of lactose intolerance (which is usual in most of the world; adult lactase persistence is the unusual condition, only common in a few populations), and it doesn't stop them. It does explain why much of the dairy that is consumed is as yogurt or other fermented milk products and cheese.