I wrote, "Už ho hledáme sedm dnů," and was told that that's wrong, and the correct answer is... Už ho hledáme sedm dnů.. And it' not the first time this has happened to me.
It is correctly entered in the system. What did the error message state? If you are sure you entered it correctly, it must be a general Duolingo problem, not a Czech course specific one.
I reversed the words hledame and ho. Uz hledame ho sedm dnu with appropriate accents. Is this wrong or just awkward?
This is a second position rule issue for ho. An accepted alternative to your answer would be "Hledáme ho už sedm dnů."
Assuming you can also say, "Už jsme ho hledali sedm dnů." what does it mean? "We'd already been looking for him for seven days."?
yes. in your czech sentence they are not looking for him anymore.