Да gives more expression to your phrase. Да нет же( "you didn't understand, what I told you") :)

Да знаю я! (No needed to talk more about that, I've understand)

Да хватит уже! (Basta!) :)

There is a small problem in Russian... we have some things that are hard to explain)))

For example, оставаться is to stay and останавливаться is to stop, but we also say останавливаться в гостинице and this means to stay at the hotel =)

I think you should practice more in conversations to understand  all this shades, it will be hard for you to understand when you only learn it here or in books.

Да нет же you will use in some dialog, when your friend is sure about something and you don't want to agree with her:

- Она была сегодня в красном платье, я точно помню!

- Да нет же! Её платье было синего цвета!

Well, I am not this good at the theory... But -ВАТЬСЯ implies the verb is in continuous or simple tense. While -ТЬСЯ makes it closer to perfect tense. Thus оставаться - to be staying or to stay, останавливаться - to be stopping or to stop. Остаться - to have stayed, остановиться - to have stopped.

As for ДА НЕТ ЖЕ - ДА is a preposition here and means BUT or AND. The phrase means DEFINETELY NOT. Or NO FOR SURE. ЖЕ enforces the expression. Had it been without ЖЕ, it would mean PROBABLY NOT.