Where are you locating, alitown?
No need to register with a practice. Depending on the doctor, you may need to call ahead and get on the list for the day. You are more likely to find that with the specialists in Santo Domingo than with doctors elsewhere. Few doctors give appointments for a specific time. This is because a great many Dominicans have difficulty being on time. So those that require you to call ahead put you on list for day, then see patients on the list in order of arrival. Doctors that don't require calling ahead see patients in order of arrival.
There is no requirement that you have one doctor for adults and another for children, and depending on where you are locating, this may not be an option. But where there is a pediatrician, many people prefer using him/her for the children, just as is also true in the US and many other countries.
The best doctors as a general rule are in Santo Domingo and Santiago. I say general rule because you can find some excellent doctors in other locations. My wife and I live in Sosua. We use area doctors for general things, but for the once or twice yearly visits to specialists, we go to Santo Domingo. Actually Santiago closer, but we lived in Samana for many years and established relationships with SD then. We do have one specialist in Puerto Plata, however, that we believe is as good as one would find anywhere in her field.
Sorry that I can't help with medical insurance questions, but there are a number of regular posters who can. I would suggest that you post your insurance questions as a new thread to increase chances of getting answers. The thread you have posted in is an old one so may not be looked at by many who might help with you question re insurance.
If you indicate where you will be locating, you may be able to get specific suggestions regarding doctors in that area.