Welcome to English Forums, Jamie. Thanks for joining us!
I have no idea what all these symbols and abreviations stand for, but good luck on your test.
"People" is the simple subject and is a noun.
"Many" is an adjective modifying people, making the complete subject "Many people."
"Are" is the being verb, requiring a "complement."
"Desperate" is the adjective complement.
"For a roof" is a prepositional phrase (adjectival) modifying "desperate."
"Over their heads" is a prepositional phrase (adjectival) modifying "roof."
That's about all I can say on the subject. If any of it is useful, I'm glad. 
Best wishes, - A.