Seven days is the point at which Tunisia stops being a highlights trip. It is enough for Carthage and the Tunis medina, the Roman cities of the interior, Kairouan, El Jem, and the Sahara with a night in the south — without spending two of the seven days recovering from the driving. Tunis to Douz is about 426 km, so a week is what lets you cross the country and come back at a sane pace.
For the north, the Roman interior and the Sahara, yes, comfortably. It is not enough to add Djerba and Tabarka as well without cutting something.
North to south and back: Tunis and Carthage first, then the Roman cities inland, Kairouan and El Jem on the way down, two nights in the south for the Chott and the dunes, and the coast on the return.
No. The country is small enough to drive — around 500 km from Tunis to the desert. A private vehicle for the week costs less than the flights would and stops wherever you want on the way.
Distances are straight-line, measured between the coordinates on our map of the sites; road distances are longer. Prices are per vehicle and come from the same database the 28 tours and 1,392 transfer fares are quoted from.