Do we ever say: "His this answer is right."
No, we do not. We might say:
His answer is right
If it's obvious which answer belongs to him and (probably) he has only submitted one answer.
So, it should be: "This answer of his is right."
Yes, this is correct.
Secondly, do we use the following structure? "This answer of him is correct."
Nope, not at all.
"Him" is an object pronoun. If you couldn't use any other noun in that place it wouldn't work. You need the possessive pronoun (or a noun in the possessive form) to show that the answer belongs to the whatever.