Yes, usually as a polite question. Example: "Shall we go now?" or "What shall we do?"

Yes, but not much, in everyday speech. We might substitute "should". EX: Shall I bring you something to eat? Should I bring you something to eat?

Shall isn't used all that often, but it is important to know. It can express something that will take place in the future. For example, "By this time next week, I shall be back home." It can be used to express an obligation or requirement: "He shall answer one day for all his crimes." "The penalty shall not exceed a $1000 fine." It can also be used in informal converstation like in the other answer: "Shall we dance?" "Shall we have lunch at noon?"

and as an auxiliary word? We shall, I shall.I shall or will go tomorrow?