Is the Basmala obligatory after supplication when encountering descriptions of believers, disbelievers, and hypocrites in the Qur'an, or can one continue reading without it, and is it permissible to interrupt recitation for supplication?

Doing so is permissible and recommended during the night vigil (Tahajjud), daytime prayers, and recitation outside of prayer. There is no need to repeat the Basmalah or the Isti'adhah afterwards. However, it should not be done in obligatory prayers, especially if you are with the Imam, in order to listen to him. Perhaps the reason for this is to lighten the burden and avoid prolonging the prayer for the congregants. As for supererogatory prayers and similar matters, the issue is broad.