Can a down rod be installed on a flush mount ceiling fan?

I would say yes because I've never seen a ceiling fan that couldn't be installed on a down rod. In fact, I've never personally seen a ceiling fan that couldn't be installed either way. You can use a down rod of any length, at your discretion. You can buy rods of various lengths or have one cut to the length you want. Beyond a few inches, maybe a foot, you will have to install additional wires to allow for length of the rod.