In my opinion, "I don't think he has money" and "I think he doesn't have money" are both equally popular. It would sound most natural if you added "any" before money in both cases. "I don't presume(assume/suppose) he has money." This would be unusual to say. You may say it to clarify your opinion, perhaps if someone else thinks that you are presuming that he does have money. However, the present simple would be an unusual choice of tense. I recommend sticking with using "presume", "suppose" and "assume" in the affirmative, in the same way that you might use "think", "believe" etc.