First lets define the term 'wide' in this context. Wide = to the fullest extent. Widely, on the other hand, is not being used correctly. Widely = over a large area. In your example, the word 'wide' is an adverb because it is being used to modify the verb 'opened'. Let me know which one you think sounds right. "He opened the door to it's fullest extent." or "He opened the door over a large area."

"He opened the door wide" is the correct way to say it.

Good question, I can see how you might be confused.