I know that Chinese Emperors have era names (like Taizong, Xuanzong, Yongle, Kangxi etc etc), as well as given names, and are often referred to by modern historians using their era names.
Given that contemporary subjects and ministers would probably never dare to refer to an Emperor by their given name and will almost always refer to them simply as Emperor:
When would a Chinese Emperor either use or be referred to by their given name, if ever at all?
I can speak to the orthodox Confucian viewpoint on the usage of personal names. Consider the following passage from Liji 礼记 (Book of Rites), describing an idealized version of Zhou rituals:
君天下，曰天子...践阼临祭祀：内事曰孝王某，外事曰嗣王某.
"The ruler of Tianxia, is called Tianzi (Emperor) ... [However] as he approaches the eastern steps of the sacrificial altar: internally, he should refer to himself as 'Filial King so-and-so', externally he should refer to himself as 'Successor King so-and-so'". (曲礼下, Qu Li Xia)
Here 'so-and-so' is meant as a stand-in for the King's personal name, and the word 祭祀 refers to an ancestral/spiritual . So clearly this is meant as an exception to the standard taboos regarding name usage, in accordance with Confucian principles that one's name should only be used in front of superiors (again from Liji):
诸侯不生命
"Nobles should not be referred to by their names while living" (曲礼下）
Certainly Kings/Emperors were expected to humble themselves before ancestors, who were regarded in some sense as superiors according to the Confucian tradition.
Other than this, I was unable to find examples where official usage of royal names was sanctioned.