Mongolia
I'm from Mongolia and want to see how China can dominate Mongolia in mid 1800's. I hope you do not so much ASB'ish thing.
During mid 1800's the Mongolia was predominately Mongolians. Chinese colonization of Mongolia began around 1875's. With China in turmoil , Greater Mongolia will take advantage and organize revolution (they did it in 1911 OTL).
Well that depends on how you define Mongolia. Large-scale Chinese immigration into Eastern Inner Mongolia, Especially Chahar actually began during the 18th century and they were a large proportion (though not a majority) of the population even in 1840. Their immigration was, at least intialy, welcomed by the Mongol nobles who found them to far more taxable than their Mongol tenants. later, attitudes soured. But I would agree that they were not militarily, socially or politically dominant in 1840.
So there will definately be a period once Qing collapse appears certain where local Mongolian nobles seize control of the Six Leagues. Can they keep control? for how long? are they capable of adompting a common agenda and leadership?
And don't expect Mongolia will just come to accept Chinese rule. No chance for such thing. Mongolia hates China, it is in Mongolian blood to hate Chinese. Also Military option is not so great for Chinese.
1. And they had backing of Russians. Russians wanted to integrate Mongolia and even count N.N.Muraviev suggested to take Mongolians in Russian Empire.
2. The logistic problem and cavalry.
This depends on several factors, some of which you mentioned:
1. Russia. Muraviev was appointed governor of Eastern Siberia in 1847 with a clear agenda, both personal and court based, to expand at the expense of Russia. It took him a bout six months to whip the administration into shape. But his predeccesor was much less energetic (and much more corrupt). If the Taiping take Beijing before he takes control then Russian intervention, though likely, will be delayed and less energetic. You also need to consider Russia's own avaliable power projection and priorities in the late 1840s.
The TransBaikal Cossacks have not yet been raised, Russian population and military presence in Eastern Siberia is still low (it doubled during Muraviev's tenancy), and the Irkutsk-Tomsk road is not yet complete. Also, I think the Greater Kazakh horde hasn't accepted Russian Suzerainty yet and they are having a nasty scuffle with them.
OTL, the Russians didn't move when the Taiping launched their Northen expedition in 1853. They only took the left Bank of the Amur in 1858, when the Qing were distracted by the Second Opium war. Bear in mind that this isn't 1911- there is no Russian railway extending to Mongolia. In fact, there is no Russian railway east of Moscow!
Bottom line- While the Russians may recognize any Mongolian state and even send some Cossacks and military trainers to help the speed and effectiveness they do so in depends greatly on the timing and length of the Rebellion. The longer it goes on before the Qing are defeated the more likely the Russians (and not only the RUssians) are likely to seek advantage.
2. Mongolian unity- My impression is that at this timepoint (1840s) there really isn't any real modern Mongolian nationalism (correct me if I'm wrong) in the sense of a literate middle class which was self concsious of it's interests Vs the Qing. So any secessionist movement is going to be one of traditional nobles whom the Qing have kept divided. Will they be able to unite around a single government as they did in 1911 or will each inner Mongolian league prince go it's own way?
Who will the dominant leaders be- Qing Bannermen deserting the sinking ship(How about Sengge Rinchen?), local nobles or the Jebtsundamba Khutuktu and the Buddhist monastic establishement?
3. Logistics and cavalry. As you said any south China based rebellion is going to be short on Cavalry. And Mongolia, unlike Manchuria offers little in the way of river or sea transportation.
OTOH, OTL the Nian rebels raised a very significant Cavalry force in the yellow river valley and Sengge Rinchen had a very tough time defeating them in the field in spite of raising Inner Mongolian troops to combat the rebels.
Fact is, various factors, such as the overwhelming dominance of the Buddhist monastic establishment and the Pauperization of Mongol commoners, meant that mongol cavalry in the mid 19th century was not what it used to be under the Yuan dynasty. This trend was excaberated by Qing policy- they wanted to gradually destroy the Mongol millitary potential and they did a good job accomplishing exactly that.
So I agree that the Taiping will face a big challenge Vs the Mongols but it's not completely insurmountable- at least in Inner Mongolia.
4. Taiping priorities. Any successful Taiping rebellion is going to have three priorities in terms of the Northern borderlands
a. Eliminate any Qing rump state. That means that primary priority is going to be placed on Manchuria, rather than Mongolia. Since both use the same logistics routes if a Mongolian seccesionist state remains neutral in the QIng-Taiping struggle it may well be left alone. For a while anyway, and maybe indefinatelry if Russia invests a great deal in it's protection.
b. LAND! A greta deal of the force impelling the Taiping rebellion and motivating Taiping troops is the lack of land in CHina proper and it's avaliability in Manchuria and Mongolia. Manchuria, however, is by far the greater prize and Outer Mongolia (no offense intended) is not much of a prize.
c. Prevent a European foothold to the North. Russian Eastern Siberia simply doesn't produce enough food to support a large Russian military force. Manchuria and Mongolia do. So in the Long term, if the Taiping think they can get away with it they are going to try and detach any Mongolian-Manchurian protectorates from Russian control (though in the case of Mongolia they may be equally satisfied vassal state rather than a province).
5. Possibility of Mongol-Chinese accomodation. Depending on how the Mongolian internal political scene and on whether a strong Qing rump state survives in Manchuria to fixate Taiping attention I can see A Mongolian government or some governments (if Inner Mongolian princes and generals fail to unify and struggle for dominance) agree to become Taiping "tributary states" and surrender foreign policy to it. On the long run that will of course create tension (over immigration, military force movements, contributions to common dfense, railroads, etc),clashes and possibly a tug of war with Russia.
Bottom line:
Without giving away too many spoliers I think outer Mongolia, Outer Manchuria , the illi Basin and Kashgaria will almost certainly become Russian protectorates or independent for the short-medium term and possibly for the long term. And that's no great loss for real chinese power- only for prestige.
Southern Manchuria (Liaoning), parts of Eastern and Southern Inner Mongolia, Inner Tibet (Qinghai) and Gansu will almost certainly be incorporated after some resistance into the Taiping state.
Everything in between (Most of Inner Mongolia, Heilongjiang, Jilin, Outer Tibet, Most of Xinjiang) is something I see as possibly going either way and I'm open for discussion about it
In fact I would be most grateful for any additional information or insights you have about how Mongolia was administered under the Qing and about the internal balance of power within Mongolian society and attitudes of nobles, monks and commoners towards Russians, Qing and Han. Not being Mongolian a lot of that information if difficult for me to access