Were there cultural, military, government administration that were put in place or highlighted during the Yuan dynasty that continued into the Ming?
There are many. The earlier scholars, such as Miyazaki Ichisada, Henry Serruys, Romeyn Taylor, and Edward Dreyer, as well as Chinese scholars such as Wang Yuquan, Yu Zhijia, and more recently Li Zhi'an all approach it mainly from a political and military perspective.
Politically, the early Ming central administrative apparatus was copied from the Yuan. There was an administrative tripod consisting of the Central Secretariat, the Censorate, and the Chief Military Commission (in the Yuan it was called the Bureau of Military Affairs). This remained the case until 1380, when Zhu Yuanzhang abolished the Central Secretariat and took direct control of the bureaucracy. The Censorate was also reformed and the Chief Military Commission was separated into five regional military commissions.
Miyazaki and Dreyer also noted that the early Ming emperors preferred military officials to civil officials. This was also true during the Yuan, as the Mongols were a martial people. During Zhu Yuanzhang's reign, military officials were paid far more than their civil counterparts. Like the Mongol emperors, Zhu Yuanzhang did not hold the civil service examinations until later into his reign because he was distrustful of Confucian scholars, and he restored the examinations most reluctantly. Even Zhu Di, who is known to have patronized Confucian scholars, used them more as advisers and secretaries than actual administrators. It was only after the Tumu Crisis of 1449, when most of the military elites were killed, did civil officials finally triumph over military officials and come to dominate the bureaucracy.
The Ming court also employed a large number of Mongols as elite soldiers and military officers. These troops were paid much better than their Chinese counterparts, which led to great discontent among the Chinese officials.
The administrative divisions of China today also had their roots in the Yuan. The Ming basically kept the Yuan's administrative divisions and in some cases broken them up into smaller divisions. The usage of the character sheng 省 to describe a province was influenced by the Yuan's Branch Secretariats (xing zhongshusheng 行中書省, or xingsheng 行省 for short). Previously, the highest level of regional administration was usually a circuit (lu 路 or dao 道). Although the Ming abolished the Branch Secretariats, the sheng was retained as an administrative region and was later adopted by the Manchu Qing.
The military bears the most striking resemblance. The Yuan had different armies divided along ethnic lines, but all were hereditary. One of the armies was the Han army, composed mainly of Chinese people who lived in northern China. When they were conscripted, the Yuan often placed two to three households together to form a single unit. Within this unit, one household furnished an adult male for military service and the other households provided financial support. Troops were expected to be self-sufficient, either by farming their own land (which was exempt from the land tax) or by farming land in state-sponsored military farming colonies. When they were on campaign, the state provided them with rations, but they were responsible for their own equipment and horses. At the same time, the military households were expected to be self-replicating. They were placed on separate military household registers and only in very rare cases could a military household be reverted back to a civilian household. This method of conscription and management was adopted by the Ming and became the basis of the weisuo system and was applied throughout China on a massive scale. The Ming also adopted the Yuan system of military farming colonies.
In addition, Zhu Yuanzhang's appointment of his many sons to fiefdoms along the border and vesting them with extensive military and administrative control over their fiefs was also a legacy of Mongol rule. It was not uncommon for the Mongols to grant land to members of the imperial family and these became known as "appanages." Although Khubilai took steps to limit their powers, the princes of these appanages still maintained a semi-independent military garrison and had the power to appoint their own officials. Khubilai also frequently sent his sons as commander-in-chief of large military border garrisons to defend against dynastic challengers.
Some recent Ming scholars led by David Robinson has begun to study the cultural influences left by the Mongols, particularly on Ming state-building and court culture. To quote:
As many studies have shown, this strand of rhetoric and policy [referring to Zhu Yuanzhang's edict expelling Mongol influences and restoring native rule] did not comprehend the full complexity of the early Ming court's attitudes toward the Yuan legacy. With often only minor changes, Hongwu adopted major aspects of the Yuan institutional apparatus, including the hereditary garrison system and the hereditary occupation system in general. He attempted to absorb areas conquered by the Mongols but not previously under Chinese control, such as Yunnan, Liaodong, and even parts of Mongolia. In many edicts, he fully acknowledged the legitimacy of the Great Yuan ulus. He recruited Mongol military personnel into his government and armies and sought to continue Yuan practices that brought Korean women and eunuchs into the imperial household. Thus, the Ming dynasty was in many ways a true successor to the Great Yuan ulus. In addition to using Confucian, Buddhist, Daoist, and other native traditions, Hongwu wished to exploit the powerful Yuan legacy to legitimate his rule within Ming territories and throughout Eurasia. (Culture, Courtiers, and Competition, page 368).
It becomes clear Zhu Yuanzhang and Zhu Di built and shaped the Ming state with the Yuan in mind. Zhu Di, in particular, saw himself as a successor to the great Khubilai Khan. Indeed, there were many similarities. Both men came to the throne through illegitimate means - Khubilai fought against his younger brother Arigh Boke and Zhu Di usurped the throne from his nephew Zhu Yunwen. Both men patronized Confucian scholars and Confucian culture. Both men cultivated good relationships with the lamas of Tibetan Buddhism. Both sought to project power across the steppes. Both chose as the site of their capital the northern city of Beijing. On page 373:
Yongle also cultivated ties with Tibetan Buddhism as a way to establish a direct association between himself and Khubilai. Both were men who founded new polities, and both chose Beijing as the site of their new regimes. As a young prince, Yongle made extensive use of the former palaces of the Great Yuan ulus, palaces constructed by Khubilai little more than a century before as a way to link the steppe with the sown. Once emperor, Yongle would establish his capital on nearly exactly the same site as the former Yuan Dadu. During his decades in Beijing, Yongle came into regular contact with many who had served at the Yuan court or in Mongol armies. Yongle was not, as is often argued, chosen to garrison the former Yuan capital because he enjoyed the special favor of his father. However, the physical space and the continuity in personnel between his princely establishment and the Dadu palaces exercised a profound influence on Yongle's ambitions and attitudes. Contemporary Ming and Choson observers regularly linked the prince's ambitions with his investiture in the former Yuan capital.