Can you fly into Zagreb out of Dubrovnik or vice-versa? That would save you some backtracking. There are also one-way flights between Zagreb and Dubrovnik that aren't too expensive usually. You could even fly into Zagreb the first day and right out to Dubrovnik to start (booked on your own), and fly out of Zagreb at the end as planned. Spending three nights in Dubrovnik to get over jet lag and then renting the car isn't a bad way to start the trip.
I wouldn't spend three nights in Split unless you have specific things to do (day trip?). Three nights is probably a lot in Dubrovnik, too, unless you have day trips planned (or if that's your arrival city as I suggest above, then you can get over jet lag there). You could stop on the way north from Split more instead: towns like Sibenik and Zadar, Krka national park (near Sibenik), maybe stop on an island like Korcula on the way north between Dubrovnik and Split, etc.
I have found driving in Croatia and Slovenia to be very easy, except in a few towns like Dubrovnik itself. (Don't need a car there proper and don't want one, probably, just a hassle.) You might find a lot of high season traffic on the roads in July, however, so plan for that. You could start in Dubrovnik, take a bus or ferry to Split, and pick up the car as you leave Split; drive to Istria, Slovenia, back down to Plitvice and end in Zagreb where you drop the car.