Many (ethnic) Chinese people take Western first names and go by them in the West and/or non-Chinese language environments, so yes, this is quite common. For example, Lee Kuan Yew was Harry Lee to many British acquaintances. (Note that the surname is first here—people vary in which order they put the names of Chinese people—and there are two non-family names, which is common for Chinese in Taiwan, SG, etc., but not in the Mainland.)

Yeah, pretty much everyone I worked with in HK had an "English" name they used day to day and an official Chinese name.

Winnie the Pooh

Yes. Seen it with Koreans too, at least if their real name is difficult to pronounce for English speakers.

Yes. People unfamiliar with pinyin often mispronounce names, particularly those with an X or Q starting letter.

Ironically, my wife's name is easy for English speakers to pronounce, but unfortunately it is not spelled like how it's pronounced.