Registration - car related, done at DGII after you buy a car and pay about 8% car value in transfer tax and re-registration. Then, every year you have to pay RD$1200 or RD$2500 tax, depending on year the car was manufactured. You get a sticker you put on your windshield. This is done August-October every year. Several banks, plus DGII offices, plus online ordering. You must carry proof of registration (copy is OK) at all times. Do NOT leave original in the car, NEVER EVER, because possession of the original means having the car ownership. If car is stolen...then...
You also have to pay RD$45 revista tax (technical control) in Banco de Reservas. You get a receipt which you take to local OTTT office and get technical control done. Then you get another sticker you put on a windshield. If they have no stickers they do no technical controls and you will be advised to have the receipt on you (in the car) at all times.
Your DL is valid for as long as your tourist visa is valid. Some claim it is valid for up to 90 days. To be on the safe side, you can do immigration renewal at Migracion within the 3 days of your arrival paying the extension fee (about 15 USD) and getting an extension in your passport. In such case, your DL is valid for up to 90 days. You can also cross to Haiti and come back to "renew" your passport stamp. In most instances showing your DL is fine for a road check, but the issues may come up with insurance claim (see thread "Driving on suspended US license").
You can get insurance directly or from several insurance brokers. Full insurance is recommended, because it will cover both 3rd party and your vehicle. Also recommended is roadside assistance and Casa de Conductor (Hotel Jail for drivers) is a must. CDC means a difference in case of an accident spending days or weeks in crowded sweaty jail cell or being put up in "open space" hotel-like "restricted area" with A/C, TV, fridge, and access to barber, shops, etc. best insurance companies are Seguros Banreservas, Proseguros, Seguros BHD, Mapfre. The worst one is Pepin. If you deal via an insurance broker they will help you sort out the paperwork in case of an accident. The premium will be high (much higher than comparable insurance in the USA) and can be paid in 4 installments (by law, it can be and must be paid within 120 days of policy effective date). How you negotiate installments is up to you, if you get good broker you will not pay any interest on deferred payment. Beware, however, that any premiums due on your part will be deducted by the insurance company from the damage payout/settlement in case of an insurance claim.