They dont need to see your NI card, you just put the number on the form.
You will have to ask them - just wait until they ask for your passport and you can just bring stuff in later, or at the interview - however they need photoID so you will probably need that letter as well as a driving license or something.
No they called me a week later, then a week after that I had induction and training.
Mine didnt tell me to wear anything so I wore jeans and a smart shirt, but it varies from coop to coop