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

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.

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!

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.

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.