Grammatically, 'I have pain' is a complete English sentence. It has a subject and verb and an object, so it is a sentence. The problem is that it's not a sentence that we ever say, because it sounds odd, slightly old-fashioned, and seems to be missing something. We need to know what kind of pain you have, or where the pain is, for example 'I have a pain in my chest'.

yes, but when you say I have pain, you should complete it by saying which part of the body has pain in it like "I have pain in my upper chest". but when you say "I am in pain", it can mean physical pain or emotional pain

'He is in pain' is the correct way to say it in English. He has pain is not a complete sentence. You would need some a long the lines as, "He has a pain is his head." Hope this helps!