I know how confusing it can be when you start out, but hope your move goes well. We are a bit further up the coast from where you are going and today has been absolutely fabulous, warm sunshine, blue skies and the sea was such a rich blue. We've been able to have our patio doors open all day - unfortunately the forecast says we are due rain at the weekend but as far as I know today (Friday) should be another good day.
Anyway, back to your query. We sorted our NIE numbers out before we moved over here last May. When we bought our car in June we needed both our NIE and our empadronamiento. We sorted out registering on the padron with our local council and gave copies of both padron and NIE to the car dealer who then taxed the car for us. Where we live you certainly don't need yout residencia to buy a car. We were told we needed our padron as proof of where we lived as the car tax is charged by the local council not nationally and different rates apply depending on where you live. You need to speak to one of the car dealers where you are thinking of buying a car from to ask exactly what will be required.
As far as I am aware you only need your NIE and and proof of where you live, to open a bank account where we live but our rental agreement was sufficient for that. Again ask at the bank where you want to open an account what they need. Registering on the padron doesn't cost much and it can be a useful document to have when you are first living out here and having to arrange so much but we have found the NIE is the important one to have.
Hope that is of some help.