I would agree with that information about "say" since it is not a direct quote like "said."
Again, I am speaking specifically of when you are paraphrasing instead of directly quoting. That is when you want to have the that. Such as in this official guide question.
"The extraordinary diary of William Lyon Mackenzie King, prime minister of Canada for over twenty years, revealed that this most bland and circumspect of men was a mystic guided in both public and private life by omens, messages received at séances, and signs from heaven."
So "revealed" is like "said" and this is when you need the that.
As to your example, I would change #3 to "I believe he will come" That is the better use of "will." If he said it, I would use "would" as in #2. "He said he would come" I think that 1 and 2 might both be acceptable in this case.