To preface, I am in no way a historian, and the content below is taken primarily from my master’s thesis, which draws from real historians and scholars (sources listed below).
Interestingly, your question touches upon a hotly debated topic in certain circles of Chinese academia, that is the ethnic identity of the Manchu rulers of the Qing dynasty. Unfortunately, I do not have much knowledge on the Han/Manchu ethnic interaction at the ground level, so I will approach this question looking at the elite-level, that is the Manchu rulers and elite of the Qing dynasty. (For more information, you could look at Joseph Esherick^1 who argues that adoption of Chinese culture took place at the grassroots level).
For some quick historical background information: The Manchus, the “ethnic group” that ruled the Qing dynasty of China, were descended from the Jurchens, an ethnic group from Northeast China. Nurhaci, a Jurchen beile (a chief of sorts), would go on to unify the disjointed Jurchen tribes into a unified power, and after conquering the Korchin Mongols, began to style himself as the khan of a northern empire to rival the Ming empire in the south, which would become what is now known as the Later Jin (后金国). Nurhaci’s successor, Hong Taiji, would transcend this title by proclaiming on October 20, 1635 the formation of the Great Qing Empire (大清国) and the transformation of the Jurchen people into the Manchu (the reasoning behind this is also debated and could be a post itself, see Rawski^2, Crossley^3, Elliott^4, Huang Xintao^5, Rhoads^6). The Qing dynasty would go on to rule over a vast multi-ethnic empire that include China Proper (中原), Tibet, Mongolia, Xinjiang, parts of Southeast Asia, and other territories.
Now onto answering your question. The prevailing consensus among traditional Chinese scholars is that in order to successfully rule the multi-ethnic empire that the Qing dynasty was, the Manchu rulers over time adopted the “superior” Han/Chinese culture, or went through what is known as Sinicization (汉化). Some say this was to appease their majority-Han subject base, or to utilize the lessons/teachings from the dynastic cultural background of Han culture, but regardless, traditional Chinese scholarship believed the Manchu rulers became Chinese by the end of the Qing dynasty.
However, in the 1990s, with China’s opening up (改革开放) foreign scholars began to have better access to China’s historical documents, sources, and artifacts. From this, some foreign scholars began to look at certain occurrences, such as the Manchu language still being used in official documents even in the late Qing, and wonder whether the Manchu rulers truly were Sinicized or did they maintain their Manchu ethnic identity in the end. This line of thinking would develop into what is now known as the New Qing History “school” (NQH), which I must note, many do not believe is a true school of thought, as the NQH scholars have mostly written independent from each other and do not share a cohesive identity. Regardless, the common thinking of NQH scholars is: the Manchu rulers of the Qing dynasty were not fully sinicized and maintained parts of their Manchu ethnic identity. Most hold the belief that it was not a one-way assimilation into Han culture as traditional thinking believed, but rather a two-way acculturation that saw give and take between Manchu and Han.
So what is the basis for the NQH scholars thinking this? The watershed moment for NQH was Evelyn S. Rawski’s speech Reenvisiong the Qing^7, where she critiqued the Chinese scholar He Bingdi^8, stating, among other things, that the success of the Qing dynasty was not Sinicization, but rather the Qing rulers maintaining their Manchu ethnic characteristics in order to connect with the non-Han people they ruled over such as in Mongolia, Xinjiang, and Tibet. This line of thinking would grow to encompass topics such as globalization, early-modernism, imperialism, what is “China”, and many other topics which I could not imagine to cover in this post. But focusing on the ethnic identity theories of NQH, the NQH scholars point to ethnic signifiers such as language, clothing, religion and rituals, among others, to prove that the Manchu rulers did maintain their Manchu ethnic identity and did not fully become Chinese.
Rawski and another scholar, Pamela Crossley, in their piece Profile of the Manchu Language^9 look at the topic of the Manchu language, and point out that for almost the entirety of the Qing dynasty, official documents used both Manchu and Chinese, and some even had Manchu writings specifically forbidding the translation of the document into Chinese. 常建华 in his piece points out how military commemorations would often be multi-lingual, using Chinese, Mongolian, Tibetan, Manchu, and even Arabic. On religion, the traditional belief was that the Manchu rulers completed adopted Confucianism as the official ideology of the Qing dynasty, however, some scholars state that while they did adopt Confucianism, it was merely one pillar among many that allowed them to interact with their multi-ethnic subjects. Li Aiyong states that is was merely one “face” of the emperor, and they would adapt their “beliefs” to the situation that they were in. So to the Han Chinese, they were Confucian, but to their Mongolian and Tibetan subjects, they were Tibetan Buddhists. This is closely related to the rituals the Qing rulers would perform. Angela Zito in Of Body and Brush^10 points to the Qing’s practice of the Grand Sacrifice to appease the Han subjects, whereas Di Cosmo in Manchu Shamanic Ceremonies at the Qing Court^11 looks at how they maintained the Manchu traditional shamanism rituals at the same time as the Confucian ones.
I could go on and on about different examples these scholars point to to prove their side is correct. But to answer your question about what is the legacy of this “ethnic mismatch”, at the end of the day Han Chinese culture has won out in modern China. After the fall of the Qing and the beginning of the Republican Era, many of China’s woes were put on the shoulders of the Qing dynasty, which in many ways led to discrimination against the Manchus by certain peoples, as they became the scapegoats. While China does have many ethnic minorities, with only 55 ethnic groups being “officially” recognized, Han Chinese makes up the vast majority of the population of China, and by consequence, has a immense influence of the “culture” of China.
**EDIT: I was speculating in this portion of my answer without sources to back myself up, which I apologize for. I cannot speak to all "high offices", but in relation to the National People's Congress (NPC), according to this official source, there appear to be 42 delegates of the 13th NPC listed as Manchu (满族), which would be about 1.4% of the total 2,980 delegates.
And to answer your second question about do Manchus hold a disproportionate number of high offices, they most certainly do not. Most officials in the Communist Party China (CCP) are Han Chinese, and while there are likely some Manchu officials there, they are most certainly a minority. (Someone with better knowledge of the Chinese political landscape may be better suited to answer that question).
I'm happy to provide more information (it was difficult trying to condense the key points down), but if you are interested in more reading into NQH, the core scholars to look at are Crossley, Rawski, and Elliott's pieces listed below. For a Chinese perspective, it is a bit harder to find good sources if you do not speak Chinese, but Li Aiyong's New Qing History and the Problem of “Chinese Empire” is a good start since it is translated to English and gives a good overview of NQH.
Sources:
Esherick, Joseph W., Hasan Kayali, and Eric Van Young. Empire to Nation: Historical Perspectives on the Making of the Modern World. Lanham, Md: Rowman & Littlefield, 2006.
Rawski, Evelyn S. The Last Emperors: A Social History of Qing Imperial Institutions. A Philip E. Lilienthal Book. Berkeley: Univ. of California Press, 1998.
Crossley, Pamela Kyle. A Translucent Mirror: History and Identity in Qing Imperial Ideology. Berkeley, Calif. London: University of California Press, 2002.
Elliott, Mark C. The Manchu Way: The Eight Banners and Ethnic Identity in Late Imperial China. Stanford, Calif: Stanford Univ. Press, 2006.
黄兴涛. “清朝满人的‘中国认同’——对美国‘新清史’的一种回应.” In 清代政治与国家认同, edited by 刘凤云, 董建中, and 刘文鹏, 16–34. 社会科学文献出版社, 2012.
Rhoads, Edward J. M. Manchus and Han: Ethnic Relations and Political Power in Late Qing and Early Republican China, 1861-1928. University of Washington Press, 2000.
Rawski, Evelyn S. “Presidential Address: Reenvisioning the Qing: The Significance of the Qing Period in Chinese History.” The Journal of Asian Studies 55, no. 4 (November 1996): 829–50. https://doi.org/10.2307/2646525.
Ho, Ping-Ti. “The Significance of the Ch’ing Period in Chinese History.” The Journal of Asian Studies 26, no. 2 (February 1967): 189–95. https://doi.org/10.2307/2051924.
Crossley, Pamela Kyle, and Evelyn S. Rawski. “A Profile of The Manchu Language in Ch’ing History.” Harvard Journal of Asiatic Studies 53, no. 1 (June 1993): 63–102. https://doi.org/10.2307/2719468.
Zito, Angela. Of Body & Brush: Grand Sacrifice as Text/Performance in Eighteenth-Century China. Chicago: University of Chicago Press, 1997.
Di Cosmo, Nicola. “Manchu Shamanic Ceremonies at the Qing Court.” In State and Ritual in China, edited by Joseph P. McDermott. Cambridge University Press, 1999.
It is worth beginning by making a few clarifications.
Firstly, the extent to which the Manchus – specifically, the Manchus as a group – ruled China is open to a considerable amount of interpretive variation. While the Qing emperors of the Aisin Gioro clan were ethnic Manchus by patrilineage and identified themselves to the Manchus as their basically divinely-appointed ruling lineage, they also mobilised other ideological components to appeal to other specific constituencies: to the Tibetans they emphasised the preceptor-donor relationship (Buddhist clerics as 'preceptors' advising temporal rulers who were 'donors' of, among other things, economic and military resources), to the Han Chinese they performed Confucian acculturation. There was definitely a component of what Mark Elliott calls 'ethnic sovereignty' to Qing rule, that is to say a consistent emphasis on the integrity of a trusted ethnic in-group to serve key military and administrative roles (organised through the Banners, themselves a multiethnic organisation in original conception), but it was always one component to Qing rule that coexisted with a broader universalism. Manchus held a privileged position in the Qing state, but they were not the Qing state as such. I think one of the better illustrations of this is in how Bannermen addressed themselves to the emperor: whereas Chinese officials referred to themselves as 'this official', Bannermen, whether writing in Chinese or in Manchu, referred to themselves as 'your slave'. The Banners could form somewhat of a distinct interest group, but they were ultimately primarily an extension of the imperial will, and not the empire's core power-brokers.
Secondly, and somewhat less importantly, Qing rule ended in 1912. The Manchu Restoration of 1917 was a brief and confused affair that was decidedly ill-advised considering what had transpired five years earlier.
So then what are the Manchu legacies in post-Qing China? Well, on the matter of societal status, it is important to note that the political privilege afforded to Bannermen under the Qing was not accompanied by significant economic privilege. Qing subsidies kept the Bannermen afloat at subsistence levels, but rarely beyond that, and so only a few major aristocratic families could sustain significant personal wealth. This was so severe in fact that, out of the negotiations over the Qing abdication in 1911, one of the concessions secured by the imperial loyalists was a continuation of the stipends for Bannermen, although these quickly broke down in the provinces and the last Banner stipends stopped being paid in 1924. Furthermore, the political prestige of Bannermen in general had declined precipitously in the last decades of Qing rule. While the Qing still concentrated on the metropolitan Banners in Beijing to a significant extent and tried to keep Manchu identity going in general, the provincial garrisons were by and large left to their own devices in the aftermath of the Taiping War, leading to the Bannermen in places like Hangzhou becoming 'orphan warriors' whose identity was tied to a state that had, seemingly, ceased to regard them as useful. It was primarily members of the imperial clan in particular, and then Beijing-based Manchus more broadly, that benefitted from the late Qing recentralisation.
The means whereby the Qing state fell further cut off the notion that the Manchus would retain any kind of unique prestige. The end of the major European monarchies was not usually based on a nationalistic rejection of a foreign aristocracy; French nobles have retained influence in France, German nobles in Germany, et cetera. But the Manchus were not Han Chinese, and, under their current designation as an ethnic minority, remain so. And the revolutionaries who overthrew the Qing state in 1911 did so not because they were wedded to republicanism in and of itself: indeed, the Qing state fell to a broad-tent coalition of republicans, constitutional monarchists, and compromisers in the imperial loyalist camp. Rather, they were very specifically opposed to Manchu rule, which they saw as deliberately negligent to Han interests, and many were inflamed by nationalist fervour buoyed by bogus race science. Three cities – Xi'an, Nanjing, and Wuchang – saw pogrom-like massacres of Manchus, with the most killings occurring at Xi'an, where 20,000 Manchus were slaughtered in the streets by fanatical Han nationalists. Had the radical republicans had their way, 1911-12 would have seen not only the expunging of Manchu political influence, but quite possibly the extermination of a substantial portion of the Manchu population, an outcome prevented in large part by key imperial loyalists, chief among them Yuan Shikai, defecting in order to bring a quick end to the violence. Self-identifying as Manchu was actively dangerous in the post-Qing period, and during the 1930s-1970s historians, sociologists and other such researchers were often unable to actually find any. Only since the post-Mao liberalisation of the PRC have Manchus been able to actually identify as such without it being a threat to their safety.
Manchus today are an ethnic minority, towards whom the state has had a variable policy. As of 2000 there was a perceptible resurgence in Manchu self-identification, something that has continued into at least the middle of the last decade, though time will tell if current PRC policy sees changes that affect that. I cannot claim anything more than some very cursory knowledge about ethnic minority policies in the post-Qing period so I won't discuss that at length, but as a minority group Manchus have not been formally afforded any kind of privilege resulting specifically from their Qing-era status. However, as of at least the 1990s they have been able to benefit to some extent from affirmative action programmes for minorities in general, such as university admissions quotas, as well as exemption from the One-Child Policy. But insofar as continued political or economic status is concerned, there has been very little Qing inheritance to serve as the basis for a considerably disproportionate political representation of Manchus. In any case, Manchus at present make up some 0.78% of the total population of the PRC, and it would take a hell of a lot of overrepresentation for them to form a considerable interest group. (For as much as it's worth, most minorities including the Manchus are somewhat over-represented in the PRC's chief legislative body, the National People's Congress: as of 2018, 44 of its 2980 delegates were Manchus (1.4%). That being said, although Han are somewhat under-represented, at 85.3% of the delegates compared to 91.6% of the national population, they are nevertheless a definite supermajority.)
That being said, we can discuss Qing legacies in post-Qing China even if Manchu legacies are less overtly apparent. Chief among them has been the territorial space we ascribe to 'China'. Regions like Xinjiang, Tibet, (Inner) Mongolia and 'the Northeast' (or 'Manchuria') that have historically fallen outside the Chinese purview formed their own integral parts of the Qing Empire, and the republic that overthrew this empire was led by elites who, like most nationalists, demanded self-determination for themselves while in the same breath insisting upon determining for others in their claimed territory, and who insisted on retaining the deposed monarchy's claimed territorial scope. This did have precedents in the Qing, under whom several regions had seen the loosening of restrictions on Han colonisation in response to military or economic pressures, or even a few particularly 'acculturated' emperors: lowland Taiwan and much of Guizhou were colonised in the 1720s, Xinjiang saw increasing Han settlement from the late 1820s and accelerating after the 1870s, outer Manchuria was opened to Han colonists in the 1850s, and the Taiwanese highlands in the 1880s. Qing (not per se Manchu) rule ended up creating many of the contours of the modern Chinese state, at least as a territorial conception, and that has perhaps proved to be its most enduring legacy.
The best overview for the topic of discourses of Manchu ethnicity in the late Qing and early Republic is Edward J.M. Rhoads' Manchus and Han: Ethnic Relations and Political Power in Late Qing and Early Republican China, 1862-1928 (2000).
Pamela Crossley's The Wobbling Pivot: China since 1800 (2010) also includes a good 3-page essay on Manchus as minorities in the book's timeframe, which informs a decent chunk of what I have written above.