Hi Claire - you will find that each area has a different set of rules about the NIE application process. It is the National Police office which issues the form, so you should enquire at your local office.
In Orihuela it can take a month to get an appointment to apply for the form! In Torrevieja you queue in the morning to get a ticket and when they have distributed a certain number of tickets they remove the machine - at least that is the system they used a few months ago. In Alicante and in Cartagena you can get them much more quickly but you need to show evidence that you need the number - if you are buying a property then you will need to have a purchase contract which you will get when you pay your deposit.
Many people either use a company to do this for them, or give a lawyer a Power of Attorney, as it is such a hassle!
As for a good bank, they are all different and tend to prefer to give mortgages on their own repossessions at the moment. It can take ages to get a decision too and they can change their mind after making you a firm offer! We always recommend using a broker as they will know the current rates and maximum amount they will lend, as well as having a relationship with the local managers.
Good luck.