You are almost right regarding the umbrellas. Oswald's father died of pneumonia after being caught in a heavy shower of rain. His mother was paranoid the same might happen to Oswald so she forced him to carry umbrella around all the time.
He got the idea to use it as a weapon when he once defended himself from bullies by physically assaulting one of them with it.
Regarding the birds, yes the name Penguin came from his appearance. But he also fell in love with birds by helping run his family's bird shop. They became his "friends" in his lonely childhood.
You know, you could make a great thread about the psychological profiles of Batman's rogues gallery. Just an idea