What is the ruling on closing the eyes when bad thoughts come to mind? Is that considered an innovation (bid‘ah)? And does it invalidate the prayer if done during it?

Closing the eyes outside of prayer is permissible. As for doing so during prayer, it does not invalidate the prayer and is not disliked if it is for the purpose of attaining khushūʿ (devotional humility) and preventing waswasah (whispers/distractions). However, it is disliked by the majority of jurists if there is no need for it, because it is an act of the Jews and a potential cause for sleep. Furthermore, the Sunnah is to cast one's gaze towards the place of prostration, and closing the eyes abandons this practice. An exception to this dislike is closing the eyes for the perfection of khushūʿ, or out of fear of losing khushūʿ, or to avoid looking at something forbidden.