What does the coral paperweight symbolize? What does Julia mean when she says, “If you kept the small rule you could break the big ones?” How does the age difference between Julia and Winston affect their beliefs? What does Winston mean when he says, “The proles are human beings. We are not human.” Is the Brotherhood real? Is O’Brien a part of the Brotherhood?

The coral paperweight symbolizes the world apart from the Party that Winston and Julia hope to construct for themselves in the room above Mr. Charrington's shop. In this world, they can live normal lives as a man and woman together in love. As Winston thinks while looking at the paperweight,

It was as though the surface of the glass had been the arch of the sky, enclosing a tiny world with its atmosphere complete. He had the feeling that he could get inside it, and that in fact he was inside it, along with the mahogany bed and the gateleg table, and the clock and the steel engraving and the paperweight itself. The paperweight was the room he was in, and the coral was Julia’s life and his own, fixed in a sort of eternity at the heart of the crystal.

When Julia says if you follow the small rules, you can break the big ones, she means that if you appear completely loyal to the state in small matters, nobody will question your loyalty. If nobody questions your loyalty, you can get away with bigger transgressions because nobody will even suspect you of them. Of course, Julia later finds out this theory is wrong.
Winston is old enough to remember a time before the Party took over Oceania. This makes him more questioning of the system than Julia. He is more quick to believe that the Party tells lies. For example, he remembers airplanes existing in an earlier time, yet the Party claims that it invented them:

Sometimes, indeed, you could put your finger on a definite lie. It was not true, for example, as was claimed in the Party history books, that the Party had invented aeroplanes. He remembered aeroplanes since his earliest childhood.

Julia, however, is young enough that all she has ever known is the Party. She is not interested in history or whether the Party tells the truth about it. She is practical and is focused on making a decent life for herself in the present.
When Winston says the proles are still human, he means that the proles can still live ordinary lives in which they care about other people, they remain loyal to one another, and they have not become hardened inside. As Winston realizes,

What mattered [in the past] were individual relationships, and a completely helpless gesture, an embrace, a tear, a word spoken to a dying man, could have value in itself. The proles, it suddenly occurred to him, had remained in this condition. They were not loyal to a party or a country or an idea, they were loyal to one another.

In contrast, Party members have become dehumanized; they are filled with hate and fear and are disconnected from caring about other people. Winston remembers that just a few weeks before, prior to Julia reconnecting him with his own humanity, he had "seen a severed hand lying on the pavement and had kicked it into the gutter as though it had been a cabbage-stalk."
The Brotherhood is most likely not real. The Party is omipotent and uses the idea of the Brotherhood to trap enemies of the state. Even if the Brotherhood were real, O'Brien would not be part of it. He believes completely in the aims of the state.