How to write Vietnamese names in English correctly? ("Việt Nam" to "Vietnam" or "Viet Nam"?)

I am from Viet Nam. In my school, when we learn English, in our books and my teacher teach us that:
1) Đà Nẵng -> Da Nang, Việt Nam -> Viet Nam
2) Hai Bà Trưng -> Hai Ba Trung
3) Bình Trị Thiên (include: Quảng Bình, Quảng Trị, Thừa Thiên - before 1976) -> Binh Tri Thien.
4) Lý Minh Nhật -> Ly Minh Nhat
5) Hồ Chí Minh -> Ho Chi Minh (but we always speak/write President Ho Chi Minh or Bác Hồ - Bac Ho - like a member in our families)
6) Nguyễn Tấn Dũng - Nguyen Tan Dung.

My English is very bad. I hope that you can understand things I write!

Nguyễn Ngọc Hà and others have stated the case for omitting all diacritics when spelling Vietnamese names in a non-linguistic English context. I do not see any justification for rearranging the order of name components. It is absurd to write "Chi Minh Ho" in the same way that it is absurd to write "Zedong Mao". Even monolingual English speakers should be expected to know that in some languages the family name comes first.

Thank you for reminding. I have been thinking this problem and your question made ​​me feel a little confused. I tried looking for information and found an article on this topic . It has changed my thinking. Part of the article translated into English or you can use the google translation tool. You can find it in this link: http://dotchuoinon.com/2010/10/20/m%E1%BB%99t-d%E1%BB%81-xu%E1%BA%A5t-nh%E1%BB%8F-vi%E1%BA%BFt-ti%E1%BA%BFng-anh-chinh-xac-cac-ten-vi%E1%BB%87t-nam/

Habits can be changed. No one specific regulations in order to change the name. I think it's like regular verbs and irregular verbs. If you change the order of my name, I will think that it is not me.

With regard to your question number 4, here (for what it's worth) is the style advice of The Oxford Guide to Style (2003):

Vietnamese

Vietnamese names should not be transposed: although the family name is first, the correct reference is to a person's second given name: Nguyen Vo Giap becomes General Giap. (This does not apply to Ho Chi Minh, which was a Chinese cover name rendered into Vietnamese pronunciation and spelling.)

As Erik Kowal notes in his answer, the vast majority of English treatments of Vietnamese words do not attempt to reproduce the diacritical marks from the original spellings.

I'm a native speaker. There's no universal application for the writing rules in Vietnamese, as the written way has changed time to time. To answer your questions, at least based on my and many others' perspectives:

1. As without diacritics, Vietnam, Hanoi, Danang, Nhatrang, Buonmethuot, Dalat, Daklak, etc. are totally fine, even preferred. The places' names have no issue, but the basic confusion lies on the variants of transliteration of spoken sounds to written texts due to diverse ethnic and political reasons.

2. It's up to one's convenience to write in either ways, which are publicly understood anyway, though as a seemingly general rule, a person's name like Ho Chi Minh, is better to be retained as such.

3. Either way is fine. I prefer Binhtrithien.

4. You write in the ways people find most convenient with. Not everyone is the same. Nowadays in Vietnam, people accept both ways, comparable with or without diacritics, though of course more commonly Chinese/Vietnamese way (last - mid - first). The first - last order is not an English order (for sure not originated in England), but globally except Chinese and its sphere of influence, including Vietnam. In fact, the tones and diacritics are critical parts of the Vietnamese names, as without them there would be some frustration. Think about e.g. Scandinavian or Balkan names, without diacritics they wouldn't be pronounced correctly. But as I said, each individual is different, so you may need to ask him/her what the most convenient ways are.

About people names, if you notate the Vietnamese order as (last - mid - first), the Western order shall be (mid - first - last) instead of (first - mid - last).

Better understand this with simply family name and given names: just swap the family name and given names, without changing the order inside the given names. So Vietnamese (F G1 G2) becomes (G1 G2 F) in the Western order. The call sign is usually G2. General Vo Nguyen Giap (Vietnamese order) is then General Giap for short, and General Nguyen Giap Vo in Western order (although not common because he is a famouse historical figure with his name fixed in its orginal order in Western publications).