I think Qing was no doubt a Chinese Dynasty. First, Manchuria was rightful territory of Ming Dynasty, and Nurhaci and his ancestors were Ming officials and vassals, he claimed he was once a "border guard of the emperor", which made him no more than just another warlord in Chinese history when he started the rebellion. Second, Qing court succeeded all political and cultural system of Ming, and the emperor(along with the whole royal family and entire nobel class)and his ministers speaked and used mandarin as not only official language but also mother tongue, Qing goverment considered itself as the mandate of China not mandate of a Manchuria state, Qing emperors considered themselves as Chinese but not Han-chinese, they claimed China belonged not only to the Han. Third, Qing government officially hold memorial ceremonies for Xuanyuan Huangdi(the Yellow Emperor), who is considered as founding father of Chinese civilization and common ancestor of all Chinese ethnic groups. The Qing ruler admited themselves as descendant of Yandi and Huangdi. There are many other reasons but I think these three can explain most of your questions.
It was very natural that japanese and koreans of that time considered Qing cort as barbarian due to the Confucian ideology, but the most importent thing is how Manchu rulers viewed themselves and how other Chinese viewed them, so I think the answer is pretty clear that Qing Dynasty WAS orthodox China
