No big deal about arriving that late at night. Remember Santiago is a big city. Airport taxis will be around at that late hour. Fare to the city should not be more than the usual US$25, it's about a 20 minute drive, maybe less at that time.
I would book into the Camino Real (cheaper) or the Gran Almirante if you are traveling with the family. They will be open because they operate a casino on the premises.
Spend two nights there so you can recover. Pick up a good map in the Nacional supermarket in Santiago, right across from the Gran Almirante Hotel. In Santiago take time to visit the Centro Leon (see
http://www.centroleon.org.do)
Then head out to Jarabacoa (land of river rafting at Rancho Baiguate -- I think they don't allow those under 14 to do rafting, but you should double check) and Constanza. Lots of great places to stay in Constanza (Alto Cerro, Rancho Constanza, etc. check them out when you get there). Don't think advance reservations are necessary, as there should be several hotels with space available.
If you have a 4 wheel car or a pick up, something sturdy, you can go from Jarabacoa to Constanza. Otherwise, come out to the Duarte Highway and then go south to take the road to Constanza. From Constanza you then take the road back to Duarte highway and from there head to San Francisco de Macoris and then to Samana. Unless you are visiting someone particular in Nagua, there is no reason to go there. Better Samana.
Don't know what you will be doing in San Juan de la Maguana, and forget about driving from San Juan to Nagua -- there is no direct route. Just do Constanza and then San Francisco de Macoris, Samana.
For a car rental, check out
http://www.dr1.com/national that has car pick up locations at the Gran Almirante Hotel and at the Santiago airport or the car rental options listed in our directory page at
http://dr1.com/directories/index.shtml
Driving is fun as long as you heed my advice to ask when you think you are wrong, and ask when you think you are headed in the right direction. Just always ask, ask, ask and ask. Have someone point out your route on your map and ask all the time. Dominicans are friendly.
Now that you will have 3 weeks time, why don't you head to Puerto Plata after visiting Constanza and Jarabacoa. Just take the road to the north. There you can visit Ocean World Park in Cofresi (see advertisement on this website and in directory listing), Columbus Water Park and Fun City for go kart racing in the afternoon. See directory listings for Puerto Plata Village, Playa Naco and Fun Royale hotels in Playa Dorada. Spend two or three nights there and then head out to visit the beach and town of Sosua. Lots of great places to stay in nearby Cabarete, which also has a great beach your family will enjoy. See hotel options in our directory page section.
And then continue on the coastal northern road to visit Playa Grande beach prior to then heading to Samana this time via Nagua (not San Francisco de Macoris). The highway goes east all the time, but you still need to ask and ask. When you get to Sanchez you have the option to turn north to Las Terrenas, or go around the peninsula to Samana city and Las Galeras.
That's what I would do. By the way, I have sons aged 16, 13 and 11, so I know what boys like!