A NIE (Número de Identidad de Extranjero) is your identification number in Spain. The Immigration service issues you the number if you want to do certain actions in Spain like renting an appartment, buying stocks, buying a property, etc.
Yes you do. Most of the economical, social or professional situations you can get involved in Spain will require the obtention of a NIE number.
This is one of the most common mistakes. The NIE number does not allow you to live in Spain.
If you are from a EU state or any other EEE state you can stay in Spain for 3 months, but after that period you will need to get the EU citizenship certificate.
If you are from any other country you will need to get a visa or the corresponding residence permit.