Poetic licence, I think - It is from the Christmas carol of the same name, where Dawel Nos usually has both words capitalised. The mutation tawel -> dawel here has a cause beyond the scope of this course, I suspect - there is nothing explicit in front of dawel to cause the mutation.
You are right, though, nos dawel 'a quiet/silent night' would be the usual word order and mutation.
It is specific to the carol 'Silent Night', and an unusual construction - 'poetic licence', if you will.
Stick with 'nos dawel' for 'a quiet night', and stick with responding 'silent night' if the sentence comes up again in Duo.
See the previous point. This is from a particular Christmas carol, not something used in day-to-day Welsh.