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:
As without diacritics, VietnamVietnam, HanoiHanoi, DanangDanang, NhatrangNhatrang, BuonmethuotBuonmethuot, DalatDalat, DaklakDaklak, 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.
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.
Either way is fine. I prefer BinhtrithienBinhtrithien.
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 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. scandinavianScandinavian or balkanBalkan names, without diacritics they wouldntwouldn'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.