What is the ruling on crying during prayer and supplication in front of people, sometimes to the point of screaming, and do these actions have any basis in Islamic law, and how can one cry out of fear of God?

Weeping out of fear of Allah or longing to meet Him is permissible and is a characteristic of the righteous servants of Allah. It is allowed in the mosque as long as it is not wailing or lamenting, and does not disturb those praying, as Abu Bakr and Umar (may Allah be pleased with them both) used to do. As for the lack of weeping, its cause is an ailment of the heart, for the soundness of the heart leads to the soundness of the limbs, including the eye.