Short answer - No.
Long answer - It is a terribly difficult thing to say that a conquering people could have a profound effect on the development of the conquered region 700 years after the conquering was done and dusted. And that would be most difficult to do while one's mind is entangled by base descriptions of "China, Russia, Central Asia, The Middle East, and India" as "back-wards".

Let's just consider one (somewhat arbitrary, as any single measure is, but nonetheless useful) measure of development: tall buildings. Of the tallest buildings in the world, 9 out of the top 10 and 20 out of the top 25 are in the regions you listed as "backwards or developing".

So perhaps you meant to ask something slightly different. Perhaps the hidden question that better represents your query is this: Did the Mongol invasions and occupations in the EDIT: 12th and 13th (originally 11th and 12th) centuries leave their civilizations less equipped to modernize at the end of the 19th and beginning of the 20th centuries?
The answer to that, again is no. Let us look at some of the regions you listed -
China - when mongol rule began to disintegrate in the early 14th century, China was wracked with what was likely the first great wave of the bubonic plague. There is at least sufficient evidence to say that at least China had to deal with wave after wave of epidemic disease. If we control for this major difficulty, we can see that China was left in pretty decent shape otherwise. So much so, that the population, economy, and military strength of the Ming was positively brimming only thirty years after the founding of the Ming in 1368. Many are aware of the great treasure fleets of Zheng He, under Emperor Zhu Di in the early 15th century. These were simply a vanguard of Ming splendor. We therefore can see that the mongol occupation of China had little to no tangible negative effects even 50 years after they were out of power.
An argument might be made to suggest that the ongoing mongol threat to the north, on the extensive northern land border, distracted the Ming and later, the Qing from what would prove to be a significant threat on the far larger sea border, from europeans. But the Ming had little difficulty with the portuguese in the 16th century, and a nominal Ming loyalist warlord, Koxinga, took just 8 months to dispatch the Dutch from Taiwan in the 17th century. So the argument is poor. What it should illustrate to you is that change is broad and powerful, and a lot faster than your question acknowledges. Too much changed between 1200 and 1850 to isolate one causality, even one so powerful in the memory and mind as the Mongol invasions.

Try India:
Well, the Mongols didn't really invade India per se. But There is a connection between those Mongols who took over what would become the Il Khanate region of central Asia, and those who would sweep down into India to create the powerful Mughal Empire. Babar was a descendent of Ghenghis Khan of course. But his conquest of India is not really part and parcel of the Mongol invasions. Nonetheless, it is impossible to effectively link his invasion with any kind of backwardness in India, largely because, like China, India wasn't terribly backward compared to anywhere else in the world. The region was not unified in the 19th century, allowing the brits to divide and conquer much of the sub-continent, but that is not the same as backwards. And remember, when the brits were conquering India, every other place on earth was backwards compared to England.

Russia - underdeveloped in the 19th century due to difficulty connecting with the overseas trade system. Nonetheless, pretty powerful by the end of the 19th century, so this one is bad also.
Middle East - Nothing to do with mongols. If anything, being held in thrall by the british raj was most of the reason for lack of development in the 19th and 20th centuries.

So, not much more to be said here I think. Long answer - also No.

There is one region that can be blamed on the Mongols for its poverty. The Mongols destroyed the underground canals in central Asia and parts of Persia and turned what was once a fertile land into an arid semi-desert. Other parts of the world recovered, but this part of the world has taken a permanent hit. Pamela Toler, Mankind: The Story of All of Us, p. 157.