Here are grammatical questions. And your corrections, please. ：þ

This time, I studied X-Men, 2000 film.

1. "He could very well be older than you, Professor." Said Jean to Charles pointing Logan.
In this case, "very well' means "distinctly"? Am I right? It can replace each other? In meaning it basically does, but you cannot just substitute it in.  The grammar would not be right
Here are two sentences I tried to make with "very well" Correct?
>> She might very well be the one who has a prejudice against the country. Correct :)
>> He would want to be the archaeologist like you very well. This one doesn't work.  Aletrnatives: 

"He would also want to be as good of an archeologist as you." (implies that someone else has already wanted to be like the "you" in the sentence)

"He would want to be the arachaeoligist that you are" implying that he wants the same level of skill as "you"
 

2. With <in all one's years>, I made three sentences. Correction please.
- Have you ever been to Korea in all your years? Sounds a bit odd, but works.
- Have you ever seen anything like this in all your years?  Sounds very good.
- I'm quite sure in all your years you've never done it like this, have you? Well done! A complex sentence perfectly executed :)

3. I heard this phrase in this film, I've never heard this kind of questions....What does it exactly mean?
# What kind of a name is Rogue?
# What kind of a name is Wolverine?
He/she wants you to tell him/her whether your last name or first name?

No, it is basically calling out the uniqueness or oddity of the name.  It's like asking, "how is that a name at all?"


4. Correction please, I made a sentence with verb "REST".
Does it sound fine to you?
- "The island rests about 54 miles from shores of Jeju-do.

It works but I would say "the island <em>lies </em>about 54 miles" instead.