Answer to 1: yes, this does refer to "fate."
Answer to 2: yes, this is a commonly used phrase in English, especially when referring to something really good happening, like love, as in the case of this song. It can be used very seriously as in "this marriage was meant to be" or not very seriously as in "We found such a good parking place! It was meant to be." Alternatively, if something goes wrong, people often say "It wasn't meant to be" to try to make themselves feel better.
On the other hand the same words can mean something far more literal. "This story was meant to be very inspring" just means that the auathor
intended the story to be inspiring.