Translation:The girl's father gives them cups of coffee.
Honestly, I think that's fine. It would sound a little odd ("the girl's father" is the more common way to say it) but no one would have difficulty understanding what you meant. "The father of the girl" is 100% grammatically correct even if it sounds a little strange to a native speaker. There's even an American movie called "Father of the bride".
The father of the girl as absolutely correct, and in some cases necessary. E.g. "The father of the girl with the striped pajamas." WOULD NOT be the same as "The girl's father father with the striped pajamas." As each sentence would describe a different person wearing the pajamas. I suppose one could say "The girl with striped pajamas' father.", but in my mind, that is the clunkier option.
See my blog post about this ... at this link.
It's not an exception. They mean different things.
Yes, I wrote a blog post about it.
"A GLASS OF WATER". If it's about water, shaped/limited/collected by the glass, it's "da". If it's about the glass, shaped/formed/owned by Mr. Water, then it's "de".
Until someone posts here why it's wrong, let's say that: Looking at such pairs x of y, if it's about the lAst, it's dA.
I'm not sure I understand what you're saying here.
Da is for quantities.
This a common question so I wrote up a detailed answer here