Are Chinese characters taught in school in Vietnam?
No. At school we only learn the meanings of basic Sino-Vietnamese syllables, not the Chinese logograms. For example we learned that tử means a child or die, but most people don't know that those are different characters (死, 子 and many more), and of course they don't know how those characters are written in Chinese.

As I know before the unification a few characters are actually taught in high school but they're not taught anymore. There are some proposals to introduce about 100 basic Chinese characters to the syllabus again but most people go against it.

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Do most Koreans and Vietnamese know the hanja/hán tự for their own names?

Do Vietnamese still use Hán tự (HanZi/ Hanja /Kanji) besides Quốc ngữ alphabet?