Both are correct! ->Shall is used less often. -- Shall is a modal. It is used to suggest an action that is likely to happen in the future.

There's instances where shall may be appropriate but overall should is much more common. Shall can be sarcastic or formal is what I am imagining. "Shall I call for your limousine, madame?" lol

Both are okay but shall is more formal and rarely used in day to day conversation in my experience.

both have the same meaning, but shall is used more than should .. that's all

Yes, in questions.