I say if you've been there more than 1 year you give two weeks. I, however, would never put what job I was going to next. They don't need to know and it's not going to add any more value to the letter.
I think this is generally true. But in my case I wanted to make it clear that I'm not just seeking a similar job elsewhere, but that I'm actually changing careers. This reinforces that there's no ill-will involved, nor am I just holding out for more money, etc. It will prevent them from trying to bug me with counter-offers or other negotiations, because there's nothing they can offer me within my new field.
I guess I could just say that "I am making a career change" and leave it at that...but there seems to be little to be gained by that minor text alteration....