"If you're going through hell, keep going." W.C

Before i begin, i can promise you my Mongol knowledge isn't as good as a lot of the people on this sub but my Great-uncle was a civil engineer who spent a lot of time in Ulaanbaatar and central Asia, he even saw a hanging in soviet occupied Afghanistan. He was also, at one point in his life, or so i am told, no stranger to Edmund Hillary. Now retired, these kind of questions is all i ask him lol - so i think my geographical understanding of the region may be up their.

The Gobi and Taklamakan are still relatively deserted, pardon the pun. Geographically speaking, deserts were basically fish-less ship-less oceans. They're only real value was as an inhospitable buffer zone that was awkward if not impossible to traverse, kind of like a mountain range or perennial Russian winter. The attrition would have been a sufficient deterrent, that's not to say the Mongols couldn't traverse the Gobi, they could, and did, as your question rightly implies. Of-course this 'deterrent' aids/hinders both the Chinese dynasties and the Mongols as each will struggle in invading the other.

That's not to say it aided/hindered them both equally. China has long been a more introverted culture, whereas the mongols were explosively not so. Also with their emphasis on shock cavalry they were far more likely to cross such a frontier. The Chinese dynasties new this of-course, that's why they built the additional protection of all those great walls on their side of the Gobi. The walls were the border and the dynasties obviously didn't seek northward expansion. If they did, they would not have hemmed themselves in like that. The Mongol's greater, more explosive range, and china's defensive attitude was why the Mongols exercised a de-facto control over the Gobi. As far as i am aware they didn't have to conquer any great desert empire, or anyone really to gain the Gobi.

My point is to emphasize that the Mongols had no great affinity for the desert, hence the Mongolian Death Worm! I think its fair to say even the Horde were a little scared of the Gobi.

If you look at this horrendously oversimplified map of Ghengis' campaigns, you'll notice the great Ghengis does largely ignore the more swollen part of the Gobi (Due south of the Merkits).

What is also obvious is that what Ghengis really excels at is swooping down and around the steppe, ravishing contemporary Russia, the Caucasus, and of-course his campaigns against the Khwarezm(ian dynasty.)

If the Gobi desert was thought to be the best option for a raid or invasion, then like the silk road traders, the horde would aim to move fast, at the right times, being as well equipped as possible, perhaps with supply trains of Bactrian camels armed to the tooth with skins of water, or perhaps gambling the even quicker 100 mile a day cavalry advances that put the fear of God into central Asia.
When this was done, they could made the likes of the desert leg of the (1211-1214) Beijing route in less than a week. Less than week on a few skins of water, your horses milk, goats meat and intestines, and whatever else you can forage/loot (often times no slim picking) is not so awful a diet/life for a 13th century mongol horseman.

Also, though i may have made it seem that way, it is woth remembering that the Gobi is not just one homogenized uninterrupted swathe of death, Parts of the Gobi are more arid than others obviously, and in some parts of it you may find surprising animals - bears, donkeys, even snow leopards. An eperianced trader/explorer/horseman would of-course know things like this. And Mongols had no shortage of them.

Often times Mongols followed trading routes, partly because traders had worked out some pretty good routes, and partly because they knew traders would frequent those routes and make easy plunder lol. Ghengis's tactic was usually to get across the Gobi quickly, where it was thin (nearer Manchuria/Mengjiang), sit just outside of spitting distance from the jin fortifications and starve them out by intercepting trade caravans and plundering the hinterlands. Its really quite genius, because you turn a beachhead on the wrong side of a treacherous frontier into something quite sustainable, thus ensuring a foothold. From there just slice at all of the main trade veins, You get richer whilst crippling and isolating them. Textbook siege. What were the Jin going to do? meet them in the field? Not a nice way to conduct a war though, with all the inevitable starvation induced cannibalism.

Of-course the best way to get at china is by sea (never get involved in a land war in Asia). As the opium war will teach you, naval bombardments can really upset china, but naval supremacy was never the Mongol's forte. If they had sea horses, forget it.

A Better land route into china is found through the likes of Urumqi and more hospitable semi mountainous passes that divide the Gobi and Taklamakan. (tian shans, down the Altun shans, more south east that divide the arid Gobi and Frozen Himalayas. This route isn't ideal by any means though, if you insist on a land invasion route into china is probably best to go down the Manchurian plain into Mengjiang trying not to stray far from the yellow sea and avoiding as much Gobi as possible.

It is largely what the Mongols did in the 13th century, and the Japanese did in the 20th, and the American led pacific coalition did in the late 21st... teehee.

Sorry i meandered, i hope i helped

Spin-off question: Did the Russian winter obstruct the Mongols at all? Or did they just shrug it off as a mild squall compared to their home steppes?

While I can't answer the question, I just want to point out many steppe confederations before (and probably after) Genghis Khan have crossed the Gobi into China.

Likewise many Chinese armies before and after him were sent north across the Gobi into the steppes.

Related questions:

How did they develop their strategy for crossing the desert? For example, did they send scouts into/across the desert to determine how much water they needed per day, and how for how many days?

Did they have outposts near or in the desert?

How did they navigate in the desert? Were there landmarks or waypoints?

If you'd like additional information, I would also recommend reading The Mongol Art of War by Timothy May. He has done extensive work over the Mongols and Genghis Khan! This work is very informative and quite fun to read!