Vietnamese culture has existed for ages.
Is the use of "for ages" correct in this sentence? In OALD, "ages" is an informal word. If so, I think the phrase "for ages" should not be used in this sentence. Please explain it to me.
Yes, that is correct. It's using the following definition of ages:
4. ages Informal An extended period of time: left ages ago.
It is equivalent to:
Vietnamese culture has existed for a very long time.
It doesn't say exactly how long they mean; in the context of a culture, I'd guess at least 1,000 years. In other contexts, it could mean almost any length of time.
Yes, it is. It's used with the persent perfect to show a period of time. (Note that we don't specifically know the exact time, we are only stating how long the situation has existed up to the present.)
The other two answers are correct, but I wanted to specifically address the "formal" part.
"Vietnamese culture has existed for ages" would be fine in nearly all contexts. It would be too informal for, say, a scholarly history of Southeastern Asia, where you would want to be more precise: "Vietnamese culture has existed for XXXX years" or perhaps "Vietnamese culture has existed since at least XXXX".