Calling one premodern regime more or less "free" than another is a bit problematic, because such questions tend to use modern metrics. However, I'll attempt to answer your individual questions to the best of my ability!

• 1. The Yuan state did link Eurasia in ways never before experienced, particularly for China. To paraphrase a classic saying, it has been said that "a man could walk from one end of the empire to the other with a golden plate, and never fear being robbed." This naturally extended the Islamic world, as the Mongols managed to conquer much of that as well. As Paul Javok Smith has written, "...the Mongol integration of China and the steppe opened geographic panoramas that would have been unimaginable within the constricted spatial confines of the Southern Song [the preceding regime]..."
• 2. Taxation in China under the Yuan functioned in complicated ways. Farmers in North China, once conquered, were directly taxed, but it seems that early rulers like Ogedei imposed limits on agricultural taxes as a conscious means of lightening fiscal burdens on the peasantry. Instead, resources were raised with the practice of "tax farming," or selling the rights to tax-collecting positions, particularly to Muslims. These tax-farmers could then potentially tax at their own rate and keep the revenue, but I don't know if we have good data or accounting to know how widespread tax-farming abuse was.
• 3. Yes and no. Chinese literati were often not considered for medium- and high-level positions. The Mongols instituted a four-tiered ethnic hierarchy, which placed Mongols at the top, followed by Central Asians, then North Chinese, and then South Chinese at the bottom. This meant "natives" were, in the state's eyes, to be considered politically inferior. Many important local administrative tasks that were once the responsibility of literati and scholars during the Song Dynasty were handed over to technical clerks, upon whom the Yuan relied heavily. In particular, South Chinese were heavily discriminated against and barred from many positions, a fact of life that southerners chafed under. That being said, many southern literati openly criticized the Yuan regime, publicly bemoaned the abandonment of the examination system, attacked Yuan administrative practices, and even found a gloating pride in the annihilation of the Mongol armada intended to invade Japan. One scholar even claimed it was Heaven's will that the invasion fail, as proof of the Mongols' ineptitude.

• We do know, however, that the combination of a more relaxed hand over southern China, plus the political stigmatization of southern Chinese, meant that populous southern regions like Jiangnan were often at the mercy of outlaws, bandit gangs, and corrupt officials/landlords/magnates.

• 4. The extent to which corruption was a problem in the Yuan as compared to other dynasties is unclear. Our knowledge of the subject is related to your sixth question (a skewed perception of the Mongol Yuan that comes from Chinese writers), which I'll get to in a moment. The Mongols were primarily concerned with ensuring no single region of the empire could amass resources and pose a threat to the state, but this was ultimately undermined by factionalism at the highest levels of court. Like I mentioned above, local administrative responsibilities were often the job of clerks. However, the darughachi system did add a layer of complexity to administration. Darughachi were Mongol statesmen paired with Chinese statesmen, and intended to oversee all administrative functions to ensure Mongol interests were fulfilled. This appears to be a more complex system, albeit one that functioned, in the sense that it tended to serve the Mongols' purposes.
• 5. Ongoing debate. Some scholars claim that the Yuan collapse resulted from mismanagement, corruption, and poor administration. Others, notably John Dardess, maintain the view that the fall of the Yuan was actually rather sudden, and resulted from extreme personal political conflicts that handicapped the ability of the state to deal with peasant insurrections in the 1350s and 60s. Specifically, the politically-motivated dismissal of the competent official Toghto is thought to have contributed to the inability of the state to respond to multiple domestic and natural crises. We don't have good demographic data for the period, but it appears the slide in population from 120 million in 1120 to 70 million by 1368 was the result of a combination of war and disease, most of which occurred during the Mongol conquests, and the late-Yuan-to-Ming unrest. It does appear that populous regions like Jiangnan experienced population increases until the late Yuan era, facilitated by large-scale, long-term dredging, diking, and draining projects that continued for decades, even centuries. These projects increased arable land and were supplemented by new agricultural techniques and tools that improved yields.
• 6. Absolutely, particularly in the modern era. I could talk about this for a loooong time (it's related to my doctorate research), but I won't, and instead merely confirm your suspicions.

Sources
Paul Javok Smith, ed. The Song-Yuan-Ming Transition in Chinese History, 2003.
Elizabeth Endicott-West, Mongolian Rule in China, 1989.
John D. Langlois, ed. China under Mongol Rule, 1981.

I'm a doctoral candidate in early Ming history, particularly the Yuan-Ming transition.

Yes, Yuan has extensive trade connections with the western Arabic world, both on land and by sea, but these connections already existed. It wasn't like when Yuan came, then all of a sudden trade routes appeared. Much like the Portuguese entry in to the Indian sea, it only added another layer of middle man you have to pay. Is this really a free-er trade system?