'Fall + ing isn’t used when fall is defined as to occur or to happen, so your first examples are incorrect.
‘The family function fell on/during that day’ is fine.
Incidentally, do you know the word ‘befall’? It is also linked to this meaning, and indicates something bad happening.
‘A terrible fate befell the couple.’
Your use of ‘fall/falling’ in these sentences sounds very wonky:
simply because it takes your mind to the first meaning of ‘fall’ suggesting ‘drop down’. It’s really only functions that fall or are falling and have the sense of ‘taking place’ as in: They didn’t like the idea of the celebration falling on the same day as the funeral.