Can what happened to the questioner's mother—the strange jolt in the car, followed by her turning off the radio which was blaring music—truly be an act of angels to alert her to the impermissibility of singing, and are angels bothered by that?

Singing with musical instruments is forbidden. The worldly punishment for it might be a mercy from God, so that the servant may repent and return to Him, as stated in His saying, the Most High: "Then We seized them with poverty and hardship so that they might humble themselves." As for linking this to a specific illness, that is a matter of the unseen which cannot be precisely interpreted. The important thing is to avoid what Allah has forbidden, repent from what one has fallen into, and not return to it.