You would say you're going to do X just to say that you're doing X. But when you say that you're going to HAVE TO do X, you're emphasizing that X is something you must do. I see this as usually being used when you must do X _instead of_ something else. For example: "I was going to help you move boxes tomorrow at 10, but I'm going to have to go to a meeting with my son's teacher at that time instead."

Yes, we use "going to have to go" often. Common examples: "I am going to have to go to the doctor tomorrow." "I'm going to have to go buy a new car battery because mine died." "I'm going to..." is just a statement that you will be going ex., "I am going to the doctor tomorrow". You could have a standing appointment or a routine physical and it has been planned for a long time. "I am going to have to go to the doctor tomorrow" expresses a need to go whether you want to or not (maybe you just sprained your ankle or a nagging cough) and most likely that it isn't pre-planned, although it could be and you might just not want to go so are lamenting that you have to go.

*explain / appreciate (typo)