The itinerary needs at least some trimming, whether you have nine or ten nights. You want an enjoyable, memorable trip, not a mad dash. I'm also thinking that you probably will need time one night to do laundry, even if it just means rinsing out some clothes in your bathroom wash basin. (You're traveling in the summer, and European cars don't have a lot of trunk space.) This will be an exhausting trip if you're on the go from 8 AM till 9 or 10 PM every night. It may be very hot in June, which will not help.
If you have just nine nights...
I'd definitely keep Plitvice. It is special, and you only need one night there. Just be sure you spend the night before your visit in or near the park and get to the park very early the next morning. It's essential to be ahead of the day-trippers. By mid- to late-morning, the walkways are unbelievable crowded, which really detracts from what should be a wonderful experience.
The first place I'd cut would be Rovinj. I love Istria, but going all the way over there (assuming you don't ferry into Pula from Venice) for a few hours in Rovinj? No; not worth it. I wouldn't go to Istria for less than three nights, to allow time for seeing at least Rovinj, Porec and either Motovun or Groznjan (or both). So include Istria only if you decide to do the Venice + Northern Croatia thing. This is an area where you'd want a car, incidentally.
I like Zadar, too. If you're willing to drop Dubrovnik (a larger and more dramatic--also more touristy--Zadar-y kind of place), you can definitely retain Zadar. Zadar is significantly more convenient, geographically. Otherwise, Zadar may need to go, depending on what other decisions you make (i.e., how much you want to do in Dubrovnik and Zagreb). You can probably see the whole Zadar historic district in 4 hours or less, but you'd have to find a place to park the car and walk in, and at some point you'll need to eat. So it would probably be pretty difficult to see Plitvice (4 or 5 hours perhaps?) and Zadar and drive to Split the same day. Incidentally, I think you'd have to make a connection if you were traveling by bus from Plitvice to Zadar.
Two nights in Split will allow you to see Split but probably not get to an island (unless that's all you want to do). That's fine; we all have to make compromises. But three nights would allow you to breathe a little and see one of the islands or Sibenik or Trogir. As far as I know, there are buses between Plitvice and Split, though I think not as frequent as the buses to and from Zagreb.
Now we come to Dubrovnik. It will probably take at least four hours to drive there from Split with no stops. There's bus service also. Two nights (just over one day) will be enough for just the old town. No side-trip to an island, Kotor or Mostar.
Now you need to turn around and head back to Zagreb (if that's your departure point). The drive will take over seven hours with no stops at all. So with one night in Zagreb you will see nothing of the city. Two nights will give you a full day for sightseeing, which I think is the minimum needed if you go there at all. It's a handsome city with an appearance different from the coastal towns. There's a lively historic district and a variety of museums, both traditional and quirky. I believe the teenagers will like the Museum of Broken Relationships, and the small museum of naïve art is impressive.
With 10 nights...
I'd still skip Istria unless you want to drop Dubrovnik.
Rovinj: skip for lack of time
Plitvice: 1 night
Zadar: skip or 1 night
Split: 2 nights (or 3 with a side trip: Sibenik, Trogir or an island)
Dubrovnik: 2 nights (or 3 with a side trip: Kotor, Mostar or an island)
Zagreb: 2 or 3 nights
Note that I have not built in dedicated beach time.
Remaing in Zagreb on arrival day may cost you a night, since you can't sleep in Zagreb and get to Plitvice early enough the next morning.