lLady = Dame versus Woman = Frau
There is a significant enough difference between "Dame" and "Frau". "Die Dame" is best translated to "lady" and "madam" whereas "Die Frau" is best translated to "woman". One is more reverential and respectful than the other.
I heard that addressing someone as Mein Herr or Meine Dame is considered a bit old-fashioned for daily conversation; is there an alternative?
Herr is Gentleman, Dame is Lady, as in "ladies and gentlemen," "meine Damen und Herren." Also, in the medieval recreation group I belong to (the Society for Creative Anachronisms) we commonly refer to each other as "my Lord" and "my Lady," which would be Herr and Dame rather than Mann and Frau (which is more like "man and wife" even though they also mean "man and woman.") At least, that is my understanding of the distinction.