You work in a very different field than I. In software engineering things work by real communication and relationships not by rules and rituals. Formality for formality's sake is frowned upon (if not openly mocked).
It is quite common for engineering jobs, even at the highest level, to be offered by email. Of course they are proceeded by the interview process, and followed up by face to face communication. There is no need for a formal ritual or for people to get bent out of shape about process. Everyone understands that what is important is the communication.
Of course, when you are leaving a company you need to notify them. An email is perfectly sufficient to let them know. No one cares about hard copy letters any more. They are just thrown away. Plus they are wasteful, and you can't search them. Email is preferable in every way.
Generally we give two weeks notice. You are right that references are important, but references come from professional relationships. Don't confuse empty formality with professional relationships, or even courtesy. The way to build up connections for personal references is to leave your work in a good state, help people with the transition, and talk to people when you leave.
I really don't see how printing a piece of paper is anything other than a meaningless anachronism. At least this is true in the software field.