Confirm with your local DR Embassy - residency visas are valid for either 60 or 90 days. You should be able to get the visa in a week or two if all your paperwork is in order. It is best to arrive here in the DR to apply for residency with as much time remaining on your residency visa as possible. DGM and many other departments sometimes view a soon to expire document as being already expired.
The residency visa allows you to enter the country for the purposes of applying for residency. The residency visa is a single entry instrument so you must submit and have your residency application accepted in Santo Domingo before you leave the country. You should be able to use the confirmation of residency application receipt issued by DGM to leave the country or prove your status if you need to.
Retain your lawyer and set up the application timetable with them before you arrive here in the DR. Your lawyer can be a big help in getting the right documents translated and smoothing over difficulties with the Embassy back home and getting some of the preliminary stuff done before you arrive in the DR.
Your lawyer will tell you how long before your visa expires they need you to be here. If you fedex the lawyer your documents, your first trip to Santo Domingo can pretty much be a get photos, fingerprints, medical exam and signature formality.
Prices vary and in most cases you get what you pay for. The ultra cheap lawyer may not be the way to go with something this important and most prices from lawyers will include all fees and associated costs, but check to make sure. Figure somewhere between $1500 - $2500 for an initial residency application. Prices at DGM do change and the category of residency you are seeking may impact the final price. The initial application is the hardest and the most important. Mucking it up may require starting the whole process again to fix it.