Is it permissible for a distressed, sorrowful Muslim to supplicate to Allah while kneeling at any time outside of prayer, and is supplication in this state more likely to be answered?

Among the recommended manners of supplication are raising the hands and facing the Qiblah, and supplication is permissible in all circumstances. As for supplicating while kneeling, there are weak narrations about it, but it is a posture that indicates submission and humility to Allah. Some of the Salaf did it in specific situations, and there is no harm in it if it is not adopted as a regular Sunnah.