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?

Yep, very well does have the meaning of distinctly but you can't alway replace them with each other. 'very well' is used more when you are saying something about someone. 
Here are two sentences I tried to make with "very well" Correct?
>> She might very well be the one who has a is prejudiced against the country.
>> He would want to be the archaeologist like you very well. (You can't use very well here. Like i said, it is usually used to indicate something about someone/thing)

Other examples: 

She could very well be his girlfriend. 

They could very well be going to dinner tonight. 

2. With <in all one's years>, I made three sentences. Correction please.
- Have you ever been to Korea in all your years?
- Have you ever seen anything like this in all your years?
- I'm quite sure that in all your years you've never done it like this, have you?

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?

Nope! When they ask something like this, it's almost like they are mocking the name. It's kind of rude as well. 
"What kind of a name is Wolverine?" ---> "Is that even a name?", "Thats a really weird name", "Why do you have such a weird name?". 


4. Correction please, I made a sentence with verb "REST".
Does it sound fine to you?

- "The island rests about 54 miles from the shores of Jeju-do. 

Hope that helps you! 

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?
Here are two sentences I tried to make with "very well" Correct?

Actually, in this case, "very well" is used more like "possibly".

"He could possibly be older than you, Professor." But, it also suggests that it is more likely to be so; maybe the word you are thinking of is definitely? "He could definitely be older that you, Professor." Either one of these would work.

>> She might very well be the one who has a prejudice against the country. 

She possibly could be...

or

She definitely could be...

>> He would want to be the an archaeologist like you very well. This use of "very well" doesn't work in this sentence. Instead you might say, "He wants to be an achaeologist like you very much."

2. With <in all one's years>, I made three sentences. Correction please.
- Have you ever been to Korea in all your years?

"Have you ever been to Korea?" or "Have you ever been to Korea in your lifetime?" (These would sound more natural).

- Have you ever seen anything like this in all your years?

Again, "Have you ever seen anything like this?" or "Have you ever seen anything like this in your lifetime?" would be more natural.

- I'm quite sure in all your years you've never done it like this, have you?

"You've never done it like this before, have you?" or "You've never done it like that in your lifetime, have you?"

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?

I agree with Sky, these are somewhat rude phrases that are mocking the person's name, as if saying it shouldn't be a name. "Why would you're name be Rogue?" or "Is Rogue even a name?"


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

Yes; rests works here because it is talking about the relationship of the distances between the two locations.

Hope this helped you!

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 lies about 54 miles" instead.