This is quite a story!Can't be of much help but here are some thoughts:
Joseph in Arabic = Youssef
Reversing of names: in Arabic names, you always inherit your father's given name as you second name.Your name, in Lebanon, would officially be Angelic Joseph Ahad.And if you had a sister named Mary, she would be Mary Joseph Ahad.Usually, the first son is named after the paternal grandfather.This explains the "reversing of names", at least as far as the first-born are concerned.Example: your paternal grandfather was named Hanna.Your father is a Joseph.If he was the first son, this means that his father's (your grandfather's) name was actually "Hanna Joseph".So now your father is Joseph (in honor of his paternal grandfather) Hanna (in honor of his father).Your older brother's name is then Hanna (in honor of your paternal grandfather) Joseph (from your father).If he follows tradition, his first son will be named Joseph Hanna. And the "reversing" continues!!
Finding relatives in Lebanon: if you have friends there, this should not be too difficult.Your friend there knows your relatives are still living.How? Can't he/she help you establish contact?
The other solution is finding your father's original documents when he arrived in the US and trace his origins back to Lebanon but if you have people there, you should try that first.
I'm not a US citizen nor do I live in the US but I find it incredible that the Social Security Office won't disclose the whereabouts of a father to his own son!