Bert:
What month of the year will you be doing this trip?
Hiking in the desert: you should check out Timna near Eilat. You should visit it on the way to the Dead Sea; why go back to Eilat? At the Dead Sea you can hike in the Ein Gedi Nature Reserve which has marked trails. If you like the American SW then you should definitely be looking at Mitzpe Ramon.
Upper Galilee: look at the National Parks web site for ideas.
Akko: one of the highlights of everybody's trip.
Let me stress again: In a three week trip you should easily be able to see ALL the regions and highlights of the country. Do one night less at Eilat, two less at the Dead Sea, one less in Jerusalem and then you will have time to visit the cost north of Tel Aviv and Mitzpe Ramon!