You're welcome, buddy. I'm sorry I can't explain it any better. Knowing something and teaching it are two very different things. Just because you know something doesn't mean you can teach it to others. :o
Yes, I think you can say that about the genitive.
And yes you can rearrange the sentence. But in your case you have to leave out "den": Die Frau gibt ihres Mannes Hut ihrem Bruder.
Other options:
Die Frau gibt den Hut ihres Mannes ihrem Bruder.
Ihrem Bruder gibt die Frau den Hut ihres Mannes.
Den Hut ihres Mannes gibt die Frau ihrem Bruder.
There are also other options, but normally nobody would use those.
We normally use the Dative first.