Commonly, in writing, the country name in Vietnamese is Việt Nam, in English is Vietnam; its capital city name in Vietnamese is Hà Nội, in English is Hanoi; its largest city name in Vietnamese is thành phố Hồ Chí Minh, in English is Ho Chi Minh city.
Should this rule apply to other normal city/province names like Đà Nẵng (to Danang?), Nha Trang (to Nhatrang?), etc?
There are place named after a person name: Hồ Chí Minh, Hai Bà Trưng. Will it be written as Ho Chi Minh city or Hochiminh city, Hai Ba Trung ward or Haibatrung ward?
There are names formed by other names, like Bình Trị Thiên is formed by Quảng Bình, Quảng Trị, Thừa Thiên. So will it be written as Binhtrithien or Binh Tri Thien?
Lastly, how should I write people names? Mostly, people will change the order of the name in Vietnamese (last – mid – first) to English order (first – mid – last), like Lý Minh Nhật to Nhat M. Ly or just Nhat Ly. As a consequence, Hồ Chí Minh should also be written as Minh C. Ho or Minh Ho. Why don’t people do that? What about not-that-much-well-known but important person like the current president? Nguyen Tan Dung or Dung Nguyen? And should I keep the tones of the names or subtract them (Nhat Ly or Nhật Lý)?
I want to know the answer from the view point of a native speakers.
You can certainly assume that English speakers will omit the tone-denoting diacritics in the Vietnamese versions of the names of people and places — partly because they don’t understand what they signify, and partly because they would have no idea how to reproduce them even if they wanted to — and that most of them will be confused about the different conventions regarding the order of names in conventional Vietnamese versus English usage.
However, the name Ho Chi Minh is so well-known to speakers of English that it has become fixed in that form, and is therefore probably immune to the reordering of its elements.
Where Vietnamese place names consist of several discrete elements (e.g. Việt Nam and Hà Nội), it seems to me that English speakers prefer to run them together, as you have already observed.
I suspect the greatest influence on all aspects of how native English speakers treat Vietnamese names is the way they are presented in newspapers, although the easier access to information about Vietnamese culture that has been made possible thanks to the Internet may prompt a few English speakers to try harder to conform to at least some of the Vietnamese norms.
You might also find some of my remarks in this discussion relevant.