I've seen it used in literature, though. For example, a son has been acting pretty rebellious for a few weeks, so his dad scolds him. The son answers back rudely, to which the dad retorts, "this is MY household! While you are part of my household, YOU FOLLOW MY RULES."
In modern language, though, I find that the suffix '-hold' is often dropped, and they just say 'house.'
I agree. I feel like Duolingo likes the word household, but is very rarely used outside of an official capacity. A household is a group of people living in a single dwelling. Does Haushalt have the same meaning? I feel we would "home" if talking about the physical place, or "family" (or similar) if talking about the people in the home.
In German, you will mostly read it in a statistical analysis of income distribution, rent index or something like that. In addition to that, "den Haushalt erledigen" means doing chores, and "haushalten" means balancing your budget. Apart from that, it's the same as you describe.
Think of it as "housekeeping". For Germans, a public budget is like "housekeeping": Cleaning your carpets, doing your dishes, getting your finances in order.
Then again, why should "house" and "budget" be the same word? "Haushalt" just isn't the same as "Haus", so "this is my house" means something completely different.