U may also say "about to die"

The others have answered. He is near to death/near to death/close to death/close to dying/about to die.

I'd say "near death," but you could use the gerund as the nominalized verb form: "He is near dying" or "He is close to dying."

yes it is incorrect because die is the event (action) of death. you can say, another way you could say this expression in a similar way is, he is close to death or he is close to dying.. or he nearly died. I don't know if I am making any sense to you but maybe this link will help. http://www.vocabulary.cl/Lists/dead-death-die-died.htm

Yes, it's incorrect. If you think about it, it makes no sense grammatically. 'Near (to)' is a preposition, so it has to be followed by a noun or a gerund.