At banks you need a passport. At my bank now I only need a copy of the passport. You will also need ID at Vimenca to change money, send or receive. Caribe Express does not require it and has better exchange rate.
As far as car docuemntation, never cary a title with you in original - whoever has the original is teh owner. Copy is widely accepted. Insurance should be original. Anyhow, in rental cars you will get only copies of everything. Drivers License, carry the original, but show the copy. You can still show the original if necessary.