Is crying sometimes during prayer, or merely making supplication sometimes, or experiencing humility with tears or trembling or a sense of closeness to God once or twice daily, considered evidence of weak faith?

Crying out of fear of Allah Almighty and desire for His Paradise is commendable and sought after. It is an indication of the strength and increase of faith, for the eye that cries out of fear of Allah will not be touched by the Fire. The Prophet, peace and blessings be upon him, and his Companions and the righteous used to cry out of fear of Allah. The same applies to trembling and feeling close to Allah during prayer. If crying and trembling have an effect on a Muslim's life, keeping him away from sins and making him feel Allah's watchfulness, then this is good. Otherwise, one must review oneself and one's faith.